I am looking at the following example of a continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ that is not differentiable at any $x\in \mathbb{R}$.

For $x\in [-1,1]$ we define $\phi (x)=|x|$ and then we extend $\phi$ to the whole $\mathbb{R}$ such that $\phi (x+2)=\phi (x)$.

Then the desired $f:\mathbb{R}\rightarrow \mathbb{R}$ is $$f(x)=\sum_{n=0}^{\infty}\left (\frac{3}{4}\right )^n\phi \left (4^nx\right )$$

Then $f$ is continuous and bounded on $\mathbb{R}$.

For...

Weierstrass function]]>

Let $I$ an interval and $f: I \to \mathbb{R}$ a differentiable (as many times as we want) function.

If $\xi \in I$ with $f''(\xi) \neq 0$, then there are $a,b \in I: a< \xi <b$ and $\frac{f(b)-f(a)}{b-a}=f'(\xi)$.

Hint: First suppose that $f'(\xi)=0$ and show that at $\xi$ we have a local extremum. Then find (for example with the intermediate value theorem) $a,b$ with $a< \xi<b$ and $f(a)=f(b)$, therefore... etc.

For $f'(\xi) \neq 0$, consider the function...

Show existence of a,b]]>

Is the situation the same for open sets or can there be sets of two, three ... elements ... ?

If there can be two, three ... elements ... how would we prove that they exist ... ?

Essentially, given the metric or distance function, I am struggling to see how in forming a set of the union of two (or more) singleton sets you can avoid including other elements of the space ...

Peter]]>

I need help in order to fully understand Example 2.7(a) ... ..

The relevant text reads as follows:

My questions are as follows:

In the above example from Willard we read the following;

" ... ... If \(\displaystyle A\) is an open...

Open Sets in R ... ... Willard, Example 2.7 (a) ... ...]]>

Could you give me a hint how to prove the following statements?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable (or twice differentiable).

- $\left.\begin{matrix}

\displaystyle{\lim_{x\rightarrow +\infty}f(x)=\ell} \ (\text{or } \displaystyle{\lim_{x\rightarrow -\infty}f(x)=\ell}), \ell\in \mathbb{R} \\

\text{and } f \text{ convex (or concave)}

\end{matrix}\right\}\lim\limits_{\substack{x\rightarrow +\infty \\ (\text{or } x\rightarrow...

Limits & properties]]>

Could you give me a hint how to explain this example?

Need help to prove statement in red frame.

Example from book (Topics In Banach Space Integration)

by Ye Guoju، Schwabik Stefan

Thank you]]>

For the functions $f:A\rightarrow B$, $g:B\rightarrow A$ and $h:B\rightarrow A$ it holds that $(g\circ f)(x)=x, \ \forall x\in A$ and $(f\circ h)(x)=x, \ \forall x\in B$. Show that it holds that $g\equiv h$.

I don't really have an idea how to show that.

Let $x\in A$. Then $(g\circ f)(x)=x \Rightarrow g(f(x))=x$. Then $f(x)\in B$.

We have that $(f\circ h)(x)=x, \ \forall x\in B$. Therefore $f(x)=(f\circ h)(f(x))$.

Is this the correct way to start? ]]>

L -3e^{9t}+9 sin(9t)

L-3e^{9t}+L 9 sin (9t)

-3 Le^{9t}+9 L sin(9t)

-3 (1/s-9) +9 (9/(s^2+9^2))

-3 (1/s-9) +9 (9/(s^2+81))

into a math program

-3*(1/(s-9)+9*(9/s exp +81)]]>

I need help to clarify an aspect of the proof of Theorem 4.2.1, the Intermediate Value Theorem ... ...

Theorem 4.2.1 and its related Corollary read as follows:

In the above proof by Silva, we read the following:

" ... ... So there exists \(\displaystyle x\) with \(\displaystyle b \gt x \gt...\)

Intermediate Value Theorem ...Silva, Theorem 4.2.1 ... ...]]>

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Theorem 3.16 ...

Theorem 3.16 and its proof read as follows:

In the above proof by Andrew Browder we read the following:

" ... ...

Intermediate Value Theorem ... Browder, Theorem 3.16 ... ...]]>

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Proposition 3.14 ...

Proposition 3.14 and its proof read as follows:

In the above proof by Browder we read the following:

" ... ... For any \(\displaystyle d \in I, d\) not an...

Increasing Function and Discontinuities ... Browder, Proposition 3.14 ... ...]]>

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help with the proof of Corollary 3.13 ...

Corollary 3.13 reads as follows:

Can someone help me to prove that if \(\displaystyle f\) is continuous then \(\displaystyle f^+ = \text{max} (f, 0)\) is continuous ...

My...

Continuity of f^+ ... Browder Corollary 3.13]]>

I need help with Stoll's proof of Theorem 3.1.16

Stoll's statement of Theorem 3.1.16 and its proof reads as follows:

Can someone please help me to demonstrate a formal and rigorous proof of the following:

If \(\displaystyle U = X \cap O\) for some open subset \(\displaystyle O\) of \(\displaystyle \mathbb{R}\) ...

... then ...

... the subset \(\displaystyle U\) of \(\displaystyle X\) is open in...

Relatively Open Sets ... Stoll, Theorem 3.1.16 (a) ...]]>

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Proposition 3.12 ...

Proposition 3.12 and its proof read as follows:

In the above proof by Browder we read the following:

" ... ... Since \(\displaystyle f(I) \subset J\)...

Composition of Two Continuous Functions ... Browder, Proposition 3.12 ... ...]]>

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Proposition 3.7 ...

Proposition 3.7 and its proof read as follows:

In the above proof by Andrew Browder we read the following:

" ... ... Clearly \(\displaystyle A\leq f(t) \leq B\)...

Increasing Function on an Interval ... Browder, Proposition 3.7 ... ...]]>

I need further help with Stoll's proof of Theorem 3.1.16

Stoll's statement of Theorem 3.1.16 and its proof reads as follows:

Can someone please help me to demonstrate a formal and rigorous proof of the following:

If the subset \(\displaystyle U\) of \(\displaystyle X\) is open in \(\displaystyle X\) ...

... then ...

\(\displaystyle U = X \cap O\) for some open subset \(\displaystyle O\) of...

Relatively Open Sets ... Another Queston Regarding Stoll, Theorem 3.1.16 (a) ...]]>

I need help with clarifying Definition 1.7.1 ...

Definition 1.7.1 reads as follows:

My question is as follows:

Is the above definition clear and correct? Is it usual?

It seems to me Conway has defined continuity at any point \(\displaystyle x \in X\) ... so...

Continuity of a Function ... Conway, Definition 1.7.1 ... ...]]>

I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ...

I need some help in understanding the proof of Theorem 5.12 ...

Theorem 5.12 and its proof read as follows:

In the above proof by Andrew Browder we read the following:

" ... ... [For instance, one can choose a positive integer \(\displaystyle n\)...

Riemann Integration .. Existence Result ... Browder, Theorem 5.12 ...]]>

I need help with an aspect of the proof of Proposition 2.1.2 ...

Proposition 2.1.2 and its proof read as follows:

In the above proof by Conway we read the following:

" ... ... Now assume that \(\displaystyle f(a_n) \to L\) whenever \(\displaystyle \{ a_n \}\) is a sequence in...

Limits of Functions ...Conway, Proposition 2.1.2 ... ...]]>

I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ...

I need some help in understanding the proof of Theorem 5.10 ...

Theorem 5.10 and its proof read as follows:

At the start of the above proof by Browder we read the following:

" ... ... The necessity of the...

The Riemann and Darboux Integrals ... Browder, Theorem 5.10 ... ...]]>

I need help with an aspect of the proof of Proposition 3.1.4 ...

Proposition 3.1.4 and its proof read as follows:

In the above proof by John Conway we read the following:

" ... ... Since \(\displaystyle \epsilon\) was arbitrary we...

The Riemann Integral ... Conway, Proposition 3.1.4 ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some help with fully understanding some remarks by Browder made after the proof of Proposition 8.21 ... ...

Proposition 8.21 (including some preliminary material and some remarks after the proof) reads as follows:

...

Differentials and Jacobians ... Remarks After Browder Prposition 8.21 ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some help with an example based only on Proposition 8.12 ... and the material in Section 8.2 preliminary to Proposition 8.12 (see scanned text at end of post) ... namely Definitions 8.9 and 8.10 and Proposition 8.11 ...

Proposition 8.12 reads as follows...

An Example on Differentials ... Browder Proposition 8.12 ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some help in formulating a proof of Proposition 8.12 ...

Proposition 8.12 reads as follows:

Can someone please help me to demonstrate a formal and rigorous proof of Proposition 8.12 using on the definitions and propositions preceding...

Differentials/Total Derivatives in R^n ... Browder, Proposition 8.12 ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in order to fully understand the proof of Theorem 8.15 ...

Theorem 8.15 and its proof read as follows:

In the above proof by Browder we read the following:

" ... ... Now...

The Chain Rule in n Dimensions ... Browder Theorem 8.15 - Another Question ...]]>

Partial fraction decomposition]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need yet further help in fully understanding the proof of Proposition 8.14 ...

Proposition 8.14 reads as follows:

In the above proof by Browder, we read the following:

" ... ... For any \(\displaystyle v \in \mathbb{R}^n\), and \(\displaystyle t \gt...\)

Yet Further Help ... Differentiabilty & Maxima ... Browder, Proposition 8.14 ... ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding some remarks by Browder made after Definition 8.9 ...

Definition 8.9 and the following remark read as follows:

In the above Remark by Browder we read the following:

"for any fixed \(\displaystyle k \neq...\)

Differentials in R^n ... Another Remark by Browder, Section 8.2 ... ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some help in order to fully understand the proof of Theorem 8.15 ...

Theorem 8.15 and its proof read as follows:

In the above proof by Browder we read the following:

" ... ... Then \(\displaystyle |k|...\)

The Chain Rule in n Dimensions ... Browder Theorem 8.15]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need yet further help in fully understanding the proof of Proposition 8.14 ...

Proposition 8.14 reads as follows:

In the above proof by Browder, we read the following:

" ... ... For any \(\displaystyle v \in \mathbb{R}^n\), and \(\displaystyle t \gt...\)

Another Question ... Differentiabilty & Maxima ... Browder, Proposition 8.14 ... ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding the proof of Proposition 8.13 ...

Proposition 8.13 reads as follows:

I think that a fully detailed proof of Proposition 8.13 reads somewhat as follows...

Differentiabilty and Continuity of Vector-Valued Functions ... Browder, Proposition 8.13 ... ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding the proof of Proposition 8.14 ...

Proposition 8.14 reads as follows:

In the above proof by Browder we read the following:

"... ... Let \(\displaystyle L = \text{df}_p\); then \(\displaystyle f(p + h) - f(p)...\)

Differentiabilty & Maxima of Vector-Valued Functions ... Browder, Proposition 8.14 ... ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding a remark by Browder after Definition 8.9 ...

The relevant text from Browder reads as follows:

At the end of the above text from Browder, we read the following:

"... ... Thus Definition 8.9...

Differentials in R^n ... Remark by Brwder, Section 8.2 ... ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need yet further help in fully understanding the proof of Proposition 8.7 ...

Proposition 8.7 and its proof reads as follows:

My question is as follows:

Can someone please demonstrate, formally and...

Operator Norm and Sequences ... Yet a Further Question ... Browder, Proposition 8.7 ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some further help in fully understanding the proof of Proposition 8.7 ...

Proposition 8.7 and its proof reads as follows:

In the above proof by Browder we read the following:

"... ... it follows...

Operator Norm and Sequences ... Another Question ... Browder, Proposition 8.7 ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding the proof of Proposition 8.7 ...

Proposition 8.7 and its proof reads as follows:

In the above proof by Browder we read the following:

"... ... Thus, \(\displaystyle \{ S_m...\)

Operator Norm and Cauchy Sequence ... Browder, Proposition 8.7 ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding the concepts in Proposition 8.6 ...

Proposition 8.6 reads as follows:

In the above proposition, Browder defines the distance function \rho (S, T) as follows:

\(\displaystyle \rho (S, T) = \| S - T \| \)

...

Operator Norm and Distance Function ... Browder, Proposition 8.6 ...]]>

Is there a specific reason why they wrote (F' * f)' instead of (f * f)' ?

I've posted a solution at the bottom of the post but I don't understand the first step: (F' * f)' = F'' * f.

Is this some standard trick that always applies?

The way I would write is \(\displaystyle f' * f = ( 2\theta(t) + \delta(t) ) *...\)

calculations with stepfunction(heaviside)]]>

I am currently reading Chapter 9: Differential Calculus in \(\displaystyle \mathbb{R}^m\) and am specifically focused on Section 9.2.1 Normed Vector Spaces of Linear Maps ...

I need some help in fully understanding Theorem 9.2.9 (3) ...

Theorem 9.2.9 (3) reads as follows:

In the proof of Theorem 9.2.9 (3) Field asserts the following...

Operator norm ... Field, Theorem 9.2.9 ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding some remarks by Browder after Lemma 8.4 pertaining to the "operator norm" Lemma 8.4 ...

The relevant text including Lemma 8.4 reads as follows:

Near the end of...

Operator norm --- Remarks by Browder After Lemma 8.4 ...]]>

I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding some remarks by Browder concerning the "operator norm" for linear transformations ...

The relevant notes form Browder read as follows...

The "Operator Norm" for Linear Transfomations ... Browder, page 179, Section 8.1, Ch. 8 ... ...]]>

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding the differences between Andrew Browder and Michael Field (Essential Real Analysis) concerning the "operator norm" for linear transformations ...

The relevant notes form Browder read as follows:

In the above...

Operator Norm ... differences between Browder and Field ...]]>

Let $X=(0,1)$. For $x,y>0$ we consider the metrices:

- $d_1(x,y)=|x-y|$
- $d'(x,y)=\left |\frac{1}{x}-\frac{1}{y}\right |=\frac{|x-y|}{|xy|}$

I want to show that these are topologically equivalent but not strongly equivalent.

$d_1$ and $d'$ are strongly equivalent iff there are constants $k>0$ and $K>0$ such that \begin{equation*}kd_1(x,y)\le d'(x,y)\le \text{ for all $x,y$.}\end{equation*}

From there we get \begin{equation*}k\le...

The metrices are topologically but not strongly equivalent]]>

Why don't mathematicians define a multiplication operation between a pair of elements and investigate the resulting field ...

For example ... why not define multiplication as X where

\(\displaystyle (x_1, x_2, \ ... \ ... \ , x_n) \ X \ (y_1, y_2, \ ... \ ... \ , y_n) = (x_1 y_1, x_2 y_2 , \ ... \ ... \ , x_n y_n)\) ...

Peter]]>

Consider the following metrices in $\mathbb{R}^2$. For $x,y\in \mathbb{R}^2$ let \begin{align*}&d_1(x,y)=|x_1-y_1|+|x_2-y_2| \\ &d_2(x,y)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2} \\ &d_{\infty}(x,y)=\max \{|x_1-y_1|,|x_2-y_2|\}\end{align*}

Draw the unit ball $B_i(0,1)=\{y\in X\mid d_i(0,y)<1\}$ in each metric.

Show that the metrices $d_1, d_2, d_{\infty}$ are strongly equivalent.

Could you give me a hint for the first part?

As for the second part:

Without loss of generality, we...

Strong equivalence of metrices]]>

Let $(X, d)$ be a metric space. For $A \subseteq X$ und $x \in X$ we define $d_A : X \rightarrow \mathbb{R}$ by \begin{equation*}d_A(x):=\inf\{d(x,y)\mid y\in A\}\end{equation*}

I want to prove the below statements:

- $A$ is closed iff for all $x\in X$ with $d(x,A)=0$ it holds that $x\in A$.
- The map $d_A:X\rightarrow \mathbb{R}$ is continuous.
- Let $A,B\subseteq X$ be disjunctive, closed subsets. Then there is a continuous function $f:X\rightarrow [0,1]$ such...

Prove statements about metric]]>

I am focused on Section 3.2 The Cauchy Riemann Equations ...

I need help in fully understanding the Proof of Theorem 3.4 ...

The start of Theorem 3.4 and its proof reads as follows:

In the above proof by Mathews and Howell we read the following:

" ... ... The partial derivatives \(\displaystyle u_x\) and...

Cauchy-Riemann Conditions for Differentiability ... Mathews & Howell, Theorem 3.4 ... ...]]>

Peter]]>

I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ...

I need help with an aspect of Example 1.5, Section 1.2, Chapter III ...

Example 1.5, Section 1.2, Chapter III, reads as follows:

At the start of the above example we read the following:

" ... ... Write \(\displaystyle \theta (z) = \text{Arg }...\)

Differentiation of the Complex Square Root Function ... ... Palka, Example 1.5, Chapter III]]>

I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ...

I have yet another question regarding Example 1.5, Section 1.2, Chapter III ...

Example 1.5, Section 1.2, Chapter III, reads as follows:

About half way through the above example from Palka we read the following:

" ... ... Since...

Yet Another Question on the Complex Square Root Function ... Palka, Example 1.5, Chapter III]]>

I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ...

I need further help with other aspects of Example 1.5, Section 1.2, Chapter III ...

Example 1.5, Section 1.2, Chapter III, reads as follows:

My questions are as follows:

In the above text by Palka we read the...

Further Questions on Complex Square Root Function ... Palka, Example 1.5, Section 1.2, Chapter III]]>

I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ...

I need help with some aspects of Examples 1.1 and 1.2, Section 1.2, Chapter III ...

Examples 1.1 and 1.2, Section 1.2, Chapter III read as follows:

My questions regarding the above two examples from Palka are as follows:

Can someone please explain...

Complex Derivatives ... Palka, Examples 1.1 and 1.2, Chapter III, Section 1.2 ...]]>

I need some help with an aspect of the proof of Theorem 7.1 ...

The statement of Theorem 7.1 reads as follows:

At the start of the above proof by Markushevich we read the following:

"If \(\displaystyle f(z)\) has a derivative \(\displaystyle f'_E(z_0)\) at \(\displaystyle z_0\), then by definition

\(\displaystyle \frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) +...\)

Diiferentiability of Functions of a Complex Variable ... Markushevich, Theorem 7.1 ... ...]]>

I am focused on Chapter 1: The Complex Plane and Elementary Functions ...

I am currently reading Chapter 1, Section 4: The Square and Square Root Functions ... and need some help in verifying a remark by Gamelin ... ...

The relevant section from Gamelin is as follows:

In the above text by Gamelin we read the following .... ...

" ... ... Every...

The Square and Square Root Functions of a Complex Variable ... Gamelin, Ch. 1, Section 4]]>

Got what seems like a simple problem at first, but has been giving me difficulty in trying to prove. Please help me with it and show me what to do because I have no clue really. Here is what it asks:

Prove that if A is a countable set and the function f:A\impliesB is onto, then B is either countable or finite.

Thanks in advance to whomever helps me out with this. The assistance is greatly appreciated.]]>

Let $K$ be a field. Suppose that there is an element $u\in K$ such that $u^2+1=0$. Show that there is no order relation on $K$ that would make $K$ an ordered field.

We have to show that no relation $<$ can exist that satisfies the order axioms, i.e.:

- Only one of $a < b$, $a = b$, or $a > b$ is true
- If $a < b$ and $b < c$, then $a < c$
- If $a < b$ and $c < d$, then $a + c < b + d$
- If $a < b$ and $c < d$, then $a c < b d$

Suppose that there is an...

Show that there is no order relation on K]]>

Let $X$ be an infinite set and let $x\in X$. Show that there exists a bijection $f:X\to X\setminus \{x\}$. Use, if needed, the axiom of choice.

To show that $f$ is bijective we have to show that it is surjective and injective.

The axiom of choice is equivalent to saying that, the function $f:X\to X\setminus \{x\}$ is surjective if and only if it has a right inverse. So we have to show that the function $f$ has a right inverse, correct?

Next we have to show that the function...

Show that there exists a bijection]]>

For each of the following functions, prove that f is one to one on E and find a formula for the inverse function f

(a) f(x)=x

(b) f(x)=((x)/(x

Please help me here, mainly with proving that each of the two functions are one-to-one, since that's what I'm mainly having trouble with. I did manage to calculate the formulas for the...

Another Question, But This One is About Proving a Function is One-to-One (PLEASE HELP)]]>

I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ...

I need help in order to fully understand a remark of Apostol in Section 16.1 ...

The particular remark reads as follows:

Could someone please demonstrate (in some detail) how it is the case that the complex function \(\displaystyle f\) has a derivative at \(\displaystyle 0\) but at no other point of...

Complex Derivative ... Remark in Apostol, Section 16.1 ... ...]]>

I'm kind of in a rush because I'll have to go to my classes soon here at USF Tampa, but I had one last problem for Intermediate Analysis that needs assistance. Thank you in advance to anyone providing it.

Question being asked: "Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x$ is an element of $A$. Prove that $\inf A=-\sup(-A)$."

This is what I had so far for a proof: "Let $\alpha =\inf A$. For any $x\in A$...

Proof of an Infimum Being Equal to the Negative Form of a Supremum (Please Help!)]]>

I have an Intermediate Analysis problem that needs assistance. I've really been having a hard time with it. This is what the question says:

"Can it happen that A⊂B (A is a subset of B) and A≠B (A does not equal B), yet sup A=sup B (the supremum of A equals the supremum of B)? If so, give an example. If not, prove why not."

Honestly, I'm not even sure where to begin with proving this, so any help would be greatly appreciated on my behalf. Thank you in advance to anyone that replies.]]>

I have this Intermediate Analysis problem that I need help finding the answer to. This is what the question asks:

"Find the supremum and infimum of each of the following sets (considered as subsets of the real numbers). If a supremum or infimum doesn’t exist, then say so. No formal proof is necessary, but give a brief justification."

This is the set in question: B={(-1)^n+((-1)^n+1)/(2n)): n is a subset of Z (the set of integers) - {0}} (meaning "not including 0).

I started out...

Infimum and Supremum of a Set (Need Help Finding Them!)]]>

The two books are as follows:

"Functions of a Complex Variable I" (Second Edition) ... by John B. Conway

"Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ...

Conway's proof that for a function of...

Complex Variable ... Differentiability Implies Continuity ... Conway and Mathews and Howell ... ...]]>

I am focused on Chapter 2: Sequences ... ...

I need help with the proof of Theorem 2.9.6 (b)

Theorem 2.9.6 reads as follows:

In the above proof of part (b) we read the following:

" ... ... Then \(\displaystyle B\) is an upper bound for every \(\displaystyle n\)-tail of \(\displaystyle \{ x_n \}\), so \(\displaystyle \overline{ x_n } = \text{sup} \{ x_k \ : \ k...\)

Upper and Lower Linits (lim sup and lim inf) - Denlinger, Theorem2.9.6 (b) ...]]>

I am focused on Chapter 2: Sequences ... ...

I need help with the proof of Theorem 2.9.6 (a)

Theorem 2.9.6 reads as follows:

In the above proof of part (a) we read the following:

" ... \(\displaystyle \forall \ m, n \in \mathbb{N}, \ \underline{x_n} \leq \underline{ x_{n + m} } \leq \overline{ x_{n + m} } \leq \overline{ x_m }\). Thus...

Another Question on Upper and Lower Limits ... Denlinger, Theorem 2.9.6 (a)]]>

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with the proof of Proposition 2.2.39 (a)

Proposition 2.2.39 (plus definitions of upper limit and lower limit ... ) reads as follows:

Can someone please demonstrate a formal and rigorous proof of Part (a) of Proposition 2.2.39 ...

Help will be appreciated ... ...

Peter]]>

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with the proof of Proposition 2.2.39 (b)

Proposition 2.2.39 (plus definitions of upper limit and lower limit ... ) reads as follows:

Can someone please demonstrate a formal and rigorous proof of Part (b) of Proposition 2.2.39 ...

Help will be appreciated ... ...

Peter]]>

I am focused on Chapter 4: Limits and Continuity ... ...

I need help in order to fully understand the example given after Theorem 4.29 ... ...

Theorem 4.29 (including its proof) and the following example read as follows:

In the Example above we read the following:

" ... ... However, \(\displaystyle f^{ -1 }\) is not continuous at...

Compact Metric Spaces and Inverse Functions ... Apostol, Example After Theorem 4.29 ...]]>

I am focused on Chapter 4: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 4.25 ... ...

Theorem 4.25 (including its proof) reads as follows:

In the above proof by Apostol we read the following:

" ... ... The sets \(\displaystyle f^{ -1 } (A)\) form an open covering of \(\displaystyle X\) ...

Functions Continuous on Comapct Sets ... Apostol, Theorem 4.25 ...]]>

I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ...

The statement of Theorem 3.1.1 (A1) reads as follows:

In the proof of Theorem 3.1.1 (A1) [see below] we read the following:

" ... ... On the other hand, with the identifications \(\displaystyle f(z) =...\)

Limits of Complex Functions ... Zill & Shanahan, Theorem 3.1.1/ A1]]>

I am focused on Chapter 3: Metrics and Norms ... ...

I need help with a remark by Carothers concerning convergent sequences in \mathbb{R}^n ...

Now ... on page 47 Carothers writes the following:

In the above text from Carothers we read the following:

" ... ... it follows that a sequence of vectors \(\displaystyle x^{ (k) } = ( x_1^k, \ ... \ ... \ , x_n^k)\) in \(\displaystyle \mathbb{R}^n\)...

The Metric Space R^n and Sequences ... Remark by Carothers, page 47 ...]]>

I am reading N. L. Carothers' book: "Real Analysis". ... ...

I am focused on Chapter 3: Metrics and Norms and Chapter 4: Open Sets and Closed Sets ... ...

I need help with an aspect of Carothers' definitions of open balls, neighborhoods and open sets ...

Now ... on page 45 Carothers defines an open ball as follows:

Then ... on page 46...

The Definition of a Neighborhood and the Definition of an Open Set ... Carothers, Chapters 3 & 4 ...]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand Example 3.10 (b) on page 95 ... ...

Example 3.10 (b) reads as follows:

My question is as follows:

Stromberg says that if \(\displaystyle X\) is any set and \(\displaystyle \mathscr{T}\) is the family of all subsets of \(\displaystyle X\) ...

... then...

The Discrete Topology and the Discrete Metric Space ... Stromberg, Example 3.10 (b) ...]]>

I am focused on Chapter 3: Metrics and Norms ... ...

I need help Exercise 32 on page 46 ... ...

Exercise 32 reads as follows:

I have not been able to make much progress ...

We have ...

\(\displaystyle B_r(x) = \{ y \in M \ : \ d(x, y) \lt r \}\)

... and ...

\(\displaystyle B_r(0) = \{ y \in M \ : \ d(0, y) \lt r \}\)

... and ...

\(\displaystyle x + B_r(0) = x + \{ y \in M \ : \ d(0, y) \lt...\)

Open Balls in a Normed Vector Space ... Carothers, Exercise 32]]>

I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ...

I need some help in order to fully understand an aspect of Proposition 6.8 ... ...

Proposition 6.8 (and the relevant Definition 6.8 ... ) read as follows:

In the above text (in the statement of Proposition 6.8 ...) we read the following:

" ... ...

Neighbourhoods and Open Neighbourhoods ... Browder, Proposition 6.8 ... ...]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.55 on page 110 ... ...

Theorem 3.55 and its proof read as follows:

At the start of the second paragraph of the above proof by Stromberg we read the following:

" ... ...Since \(\displaystyle A_1^{ - \ \circ } = \emptyset\), we can...

Baire Category Theorem ... Stromberg, Theorem 3.55 ... ...]]>

I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ...

I need some help in order to fully understand a statement by Browder in Section 6.1 ... ...

The relevant statements by Browder follow Definition 6.10 and read as follows:

In the above text we read the following...

Boundary of a Set in a Topological Space ... Browder, Remarks Following Defn 6.10, pages 125-126 ...]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need further help in order to fully understand the proof of Theorem 3.47 on page 107 ... ...

Theorem 3.47 and its proof read as follows:

In the third paragraph of the above proof by Stromberg we read the following:

" ... ... But \(\displaystyle U \cap V \cap [a, b] \ \subset \ U \cap V \cap S =...\)

Connectedness and Intervals in R ... Another Question ... Stromberg, Theorem 3.47 ... ...]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.47 on page 107 ... ...

Theorem 3.47 and its proof read as follows:

In the second paragraph of the above proof by Stromberg we read the following:

" ... ... Since \(\displaystyle U\) is open we can choose \(\displaystyle c' \gt c\) such...

Connectedness and Intervals in R ... Stromberg, Theorem 3.47 ... ...]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need

Theorem 3.43 and its proof read as follows:

At about the middle of the above proof by Stromberg we read the following:

" ... ... Otherwise...

Equivalent Statements to Compactness ... Another Question ... Stromberg, Theorem 3.43 ... ...]]>

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.43 on pages 105-106 ... ...

Theorem 3.43 and its proof read as follows:

At about the middle of the above...

Equivalent Statements to COmpactness ... Stromberg, Theorem 3.43 ... ...]]>

My mother language is not English by the way. Sorry for spelling and gramme.

I'm curious to see if you can help me with my problem.I have already tried for almost a week and did not get to a solution. I also know, that the Maximum likelihood estimation is part of statistics and probability calculation. But since it is about formula transformation, I used the analysis forum.

The Maximum likelihood estimation formula is the following:

$L(\theta = p) =...

Maximum likelihood estimation]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Lemma 3.44 on page 105 ... ...

Lemma 3.44 and its proof read as follows:

In the above proof by Stromberg we read the following:

" ... ... Also, if \(\displaystyle x \in X\) and \(\displaystyle \epsilon \gt 0\), it follows from (2) that...

Countably Dense Subsets in a Metric Space ... Stromberg, Lemma 3.44 ... ...]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.40 on page 104 ... ...

Theorem 3.40 and its proof read as follows:

In the above proof by Stromberg we read the following:

" ... ... To see that \(\displaystyle S\) is bounded, consider \(\displaystyle \mathscr{U} = \{ B_k (0) \ : \ k \in...\)

Heine-Borel Theorem in R^n ... Stromberg, Theorem 3.40 ... ...]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ...

Theorem 3.36 and its proof read as follows:

In the above proof by Stromberg we read the following:

" ... ...Next let \(\displaystyle U = \bigcap_{ k = 1 }^n U_{ y_k }\). Then \(\displaystyle U\) is a neighbourhood...

Compact Topological Spaces ... Stromberg, Theorem 3.36 ... ...]]>

following statements are equivalent:

i) limx→∞ f(x) = L

ii) For every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn))

converges to L.

Not even sure how to begin this one, other than the fact that proving i) -->ii) and ii)--> i) will be sufficient. Could anyone help me with these 2 parts?

Thanks!]]>

x→∞

Pretty intuitive when considering the graph of the function. But how would I show this using the epsilon-delta definition?

Thanks!]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand an aspect of Example 3.34 (c) on page 102 ... ...

Examples 3.34 (plus some relevant definitions ...) reads as follows:

In Example 3.34 (c) above from Stromberg we read the following:

" ... ... Let \(\displaystyle \mathscr{I}\) be the collection of all...

Compact Topological Spaces ... Stromberg, Example 3.34 (c) ... ...]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ...

Theorem 3.6 and its proof read as follows:

In the above proof by Stromberg we read the following:

" ... ... Letting \(\displaystyle r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}\) we see that \(\displaystyle B_r (a)...\)

Open Subsets in a Metric Space ... Stromberg, Theorem 3.6 .. ...]]>

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.18 on pages 98-99 ... ...

Theorem 3.18 and its proof read as follows:

In the above proof by Stromberg we read the following:

" ... ... If \(\displaystyle a_x \lt t \leq x\), then \(\displaystyle t\) is...

Structure Theorem for the Open Subsets of R... Stromberg, Theorem 3.18 ... ...]]>

proof.

I believe 0 is a cluster point but I can't figure out how to prove this, or how to prove any other point is not.

Any quick help would be appreciated. Thanks.]]>

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with an aspect of Example 2.3.52 ...

The start of Example 2.3.52 reads as follows ... ...

In the above Example from Sohrab we read the following:

" ... ... Then, given any sequences \(\displaystyle x = (x_n), \ y = (y_n) \in l^2 ( \mathbb{N} )\), the series \(\displaystyle \sum_{ n = 1 }^{...\)

The Class of All Square-Summable Real Sequences... l^2(N) ... Sohrab Example 2.3.52 ... ...]]>

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$. Is there any class of class of functions and some kind of "growth conditions" such that bounds like below can be established:

\begin{equation}

||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),

\end{equation}

with $\mathcal{X}$ := $\{x:f(x)=0\}$ (zero set of $f$) and some function $g$ (e.g. $\ell 2$-norm).

I am interested to know the class of functions. Any help will help a lot. Thanks in advance]]>

I am focused on Chapter 4: Topology of [FONT=MathJax_AMS]R[/FONT] and Continuity ... ...

I need help in order to fully understand the proof of Theorem 4.3.4 ... ...

Theorem 4.3.4 and its proof read as follows:

In the above proof by Sohrab we read the following:

" ... ... Therefore \(\displaystyle f^{ -1 } (O') = S \cap O\) for some open set \(\displaystyle O\) ... ... "...

Continuity and Open Sets ... Sohrab, Theorem 4.3.4 ... ...]]>

I am focused on Chapter 4: Topology of [FONT=MathJax_AMS]R[/FONT] and Continuity ... ...

I need help in order to fully understand the proof of Theorem 4.1.10 ... ...

Theorem 4.1.10 and its proof read as follows:

In the above proof by Sohrab we read the following:

" ... ...Since \(\displaystyle [a, b]\) is compact (by Proposition...

Heine-Borel Theorem ... Sohrab, Theorem 4.1.10 ... ...]]>