What is Real analysis: Definition and 509 Discussions
In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.
$$r<x<s$$
$$s-r>0$$
We enploy the Archimedean principle where
$$n(s-r)>1$$
We employ density of rationals where
$$\exists [m,m+1] \in Q$$
Such that
$$nr\in [m,m+1)$$
Therefore
$$m\leq nr \lt m+1$$$$ \frac m n \leq r \lt \frac m n + \frac 1 n $$
Since
$$ \frac m n \leq r $$
Then
$$...
Summary: Is Baby Rudin a good choice for first Real Analysis textbook for someone without strong pure math background?
I've completed 2 semesters of college calculus, but not "pure math" calculus which is taught to math students. I'm looking for introductory text on Real Analysis and I've...
Summary: Rudin theorem 1.21
He has said that as t=X/(X+1) then t^n<t<1 then maximum value of t is 1. then in the next part he has given that t^n<t<x. as maximum value of t is less than 1 why has he given that t<x ?
For the set A:
Note that if n is odd, then ## A = \{ -1 + \frac {2} {n} : \text{n is an odd integer} \} ## . If n is even, A = ## \{1 + ~ \frac {2} {n} : \text{ n is an even integer} \} ## .
By a previous exercise, we know that ## \frac {1} {n} ## -> 0. Let ## A_1 ## be the sequence when n...
Does anyone have a recommendation for a book(s) to use for the self-study of real analysis? I have just finished Apostol Calculus, Vol. 2 and would like to move on to real analysis. I am not sure whether I should continue following Apostol and move on to Apostol mathematical analysis or...
Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established:
\begin{equation}
||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),
\end{equation}
with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement:
$$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the...
Homework Statement
We've been given a set of hints to solve the problem below and I'm stuck on one of them
Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0
Hint
let set S = {x∈[a,b]:f(x)≤0}
let c =...
Hi forum.
I'm trying to prove a claim from Mathematical Analysis I - Zorich since some days, but I succeeded only in part.
The complete claim is:
$$\left\{\begin{matrix} f\in\mathcal{C}^{(n)}(-1,1) \\ \sup_{x\in (-1,1)}|f(x)|\leq 1 \\ |f'(0)|>\alpha _n \end{matrix}\right. \Rightarrow \exists...
Problem:
Let $\left(X, M, \mu\right)$ be a probability space. Suppose $f \in L^\infty\left(\mu\right)$ and $\left| \left| f \right| \right|_\infty > 0$. Prove that
$lim_{n \rightarrow \infty} \frac{\int_{X}^{}\left| f \right|^{n+1} \,d\mu}{\int_{X}^{}\left| f \right|^{n} \,d\mu} = \left| \left|...
Hey Guys, I posed this on Math Stackexchange but no one is offering a good answering. I though you guys might be able to help :)
https://math.stackexchange.com/questions/3049661/single-point-continuity-spivak-ch-6-q5
Homework Statement
2. Relevant equation
Below is the definition of the limit superior
The Attempt at a Solution
I tried to start by considering two cases, case 1 in which the sequence does not converge and case 2 in which the sequence converges and got stuck with the second case.
I know...
Homework Statement
Let us have ##n \geq 3## points in a square whose side length is ##1##. Prove that there exists a graph with these points such that ##G## is connected, and
$$\sum_{\{v_i,v_j\} \in E(G)}{|v_i - v_j|} \leq 10\sqrt{n}$$
Prove also the ##10## in the inequality can't be replaced...
Homework Statement
If I have the two curves
##\phi (t) = ( \cos t , \sin t ) ## with ## t \in [0, 2\pi]##
##\psi(s) = ( \sin 2s , \cos 2s ) ## with ## s \in [\frac{\pi}{4} , \frac{5 \pi}{4} ] ##
My textbook says that they are equivalent because ##\psi(s) = \phi \circ g^{-1}(s) ## where ##...
Homework Statement
Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set.
Homework Equations
n/a
The Attempt at a Solution
Let ##x \in T##, then ##\exists \sigma \in S## such that ##x...
Homework Statement
Let ##f:\Bbb{R} \to \Bbb{R}## be a function such that ##f## has a local minimum for all ##x \in \Bbb{R}## (This means that for each ##x \in \Bbb{R}## there is an ##\epsilon \gt 0## where if ##\vert x-t\vert \lt \epsilon## then ##f(x) \leq f(t)##.). Then the image of ##f## is...
I would like to learn (self-study) the theory behind the n-dimensional Riemann integral (multiple Riemann integrals, not Lebesgue integral). I am from Croatia and found lecture notes which Croatian students use but they are not suitable for self-study. The notes seem to be based on the book: J...
I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it.
Problem:
Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence $(a_n)$ of integers...
Here are a couple more problems I am working on!
Problem 1:
Prove that,
$limsupa_n+liminfb_n \leq limsup(a_n+b_n) \leq limsupa_n+limsupb_n$
Provided that the right and the left sides are not of the form $\infty - \infty$.
Proof:
Consider $(a_n)$ and $(b_n)$, sequences of real numbers...
I am working a bunch of problems for my Real Analysis course.. so I am sure there are more to come. I feel like I may have made this proof too complicated. Is it correct? And if so, is there a simpler method?
Problem:
Show that $liminfa_n \leq limsupa_n$.
Proof:
Consider a sequence of real...
In Rudin, the derivative of a function ##f: [a,b] \to \mathbb{R}## is defined as:
Let ##f## be defined (and real-valued) on ##[a,b]##. For any ##x \in [a,b]##, form the quotient ##\phi(t) = \frac{f(t) - f(x)}{t-x}\quad (a < t <b, t \neq x)## and define ##f'(x) = \lim_{t \to x} \phi(t)##, if the...
I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...
Homework Statement
Show that ##f(x)=\frac{1}{x^2}## is not uniformly continuous at ##(0,\infty)##.
Homework Equations
N/A
The Attempt at a Solution
Given ##\epsilon=1##. We want to show that we can compute for ##x## and ##y## such that ##\vert x-y\vert\lt\delta## and at the same time ##\vert...
Homework Statement
Let ##K\neq\emptyset## be a compact set in ##\Bbb{R}## and let ##c\in\Bbb{R}##. Then ##\exists a\in K## such that ##\vert c-a\vert=\inf\{\vert c-x\vert : x\in K\}##.
2. Relevant results
Any set ##K## is compact in ##\Bbb{R}## if and only if every sequence in ##K## has a...
Homework Statement
Let ##p\in\Bbb{R}##. Then the function ##f:(0,\infty)\rightarrow \Bbb{R}## defined by ##f(x):=x^p##. Then ##f## is continuous.
I need someone to check what I've done so far and I really need help finishing the last part. I am clueless as to how to show continuity for...
Hey,
I tried to construct the derivation of the integral C with respect to Y:
$$ \frac{\partial C}{\partial Y} = ? $$
$$ C = \frac{2}{\pi} \int_0^{\infty} Re(d(\alpha) \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha $$
with
$$d(\alpha) = exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z...
Homework Statement
Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero.
2. Relevant definitions and propositions:
3. The attempt at a proof:
Proof:(by construction)
Let...
Homework Statement
"A set ##A\subset [0,1]## is dense in ##[0,1]## iff every open interval that intersects ##[0,1]## contains ##x\in A##. Suppose ##f:[0,1]\rightarrow ℝ## is integrable and ##f(x) = 0,x\in A## with ##A## dense in ##[0,1]##. Show that ##\int_{0}^{1}f(x)dx=0##."
Homework...
Homework Statement
1) Suppose ##t## is a subsequential limit for ##(s_n)##. Write the precise definition of the meaning of this statement.
2) Explain why there exists a strictly increasing sequence ##(n_k)^\infty_{k=1}## of natural numbers such that ##\lim s_{n_k}=t##.
Homework EquationsThe...
This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...
Homework Statement
Suppose that ##( s_n )## and ## (t_n)## are bounded sequences. Given that ##A_k## is an upper bound for ##\{s_n : n \ge k \}## and ##B_k## is an upper bound for ##\{t_n : n \ge k \}## and that ##A_k + B_k## is an upper bound for ##\{s_n + t_n : n \ge k \}##, show that ##\sup...
Homework Statement
Question: Let ##I = [0,1]##. Suppose ##f## is a continuous mapping of ##I## into ##I##. Prove that ##f(x) = x## for at least one ##x∈I##.
Homework Equations
Define first(##[A,B]##) = ##A## and second(##[A,B]##) = ##B## where ##[A,B]## is an interval in ##R##.
The Attempt at...
Dear Everybody,
I need some help with seeing if there any logical leaps or any errors in this proves.
Corollary 1.2.8 to Proposition 1.2.8 states:
if $S\subset\Bbb{R}$ is a non-empty set, bounded from below, then for every $\varepsilon>0$ there exists a $y\in S$ such that $\inf...
Prove this Proposition 1.2.6 v in Introduction to Real Analysis by Jiri Lebl
Dear Everybody,
I need some help with seeing if there are any logical leaps or errors in this proof.
The theorem states:
$A\subset\Bbb{R}$ and $A\ne\emptyset$
If $x<0$ and A is bounded below, then...
I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ...
I am focused on Chapter 1: Continuity ... ...
I need help with an aspect of Lemma 1,1,7 (ii) ...
Duistermaat and Kolk"s Lemma 1.1.7 reads as follows:
In the above Lemma part (ii)...
Homework Statement
Show that ##\sum_{k=2}^\infty d_k## converges to ##\lim_{n\to\infty} s_{nn}##.
Homework Equations
I've included some relevant information below:
The Attempt at a Solution
So far I've managed to show that ##\sum_{k=2}^\infty |d_k|## converges, but I don't know how to move...
Hello,
Is there a difference from these courses, or are they the same course with different names? I need to know which one to choose for the upcoming semester...
Intro to Analysis, Intro to Real Analysis I, and Numerical Analysis
Thank you,
Tracie
Homework Statement
If ##\{K_\alpha\}## is a collection of compact subsets of a metric space X, such that the intersection of every finite subcollection of {##K_\alpha##} is nonempty, then ##\cap K_\alpha## is nonempty. Generalize this theorem and proof the generalization. Why doesn't it make...
Let $$f:\Omega\to\mathbb{R}$$, where $$\Omega\subset\mathbb{R}^d$$, and $$\Omega$$ is convex and bounded. Let $$\{x_i\}_{i=1,2,..N}$$ be a set of points in the interior of $$\Omega$$. $$d_i\in\mathbb{R}$,$i = 1,2,..N$$
I want to solve this weakly formulated pde:
$$
0=\frac{A}{N^{d+1}} \sum_i...
Hi, all.
I would like to read books about the topics - Geometry, Real Analysis and Electricity and Magnetism. And I find the followings. Are they decent and rigorous?
Geometry
The Real Numbers and Real Analysis
Introduction to Electrodynamics
Classical Electricity and Magnetism
Electricity...
Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##,
$$\frac{\partial}{\partial r_k}\int_V...
Dear friends,
I have found a derivation of the fact that, under the assumptions made in physics on ##\rho## (to which we can give the physical interpretation of charge density) the function defined by
$$V(\mathbf{x},t):=\frac{1}{4\pi\varepsilon_0}\int_{\mathbb{R}^3}...
Can anyone explain the behavior of light I came across as I sat in my lounge this evening having a nice cup of Mocha . Hint ( I am sitting in a room with some led ceiling lights on) can you:
1.Guess how many Led lights I have on
2.Explain the appearance of light which is looking like a typical...
Homework Statement
Give an example to show that the given "definition" of limx→aƒ(x) = L is incorrect.
Definition: For each 0<δ there is an 0<ε such that if 0< l x-a I < δ , then I ƒ(x) - L I < ε .
Homework EquationsThe Attempt at a Solution
I considered the piece-wise function: ƒ(x) = (0 if...
Hello! Can someone explain to me, in real analysis, what is the difference in expanding a function as a Taylor series around 2 different point. So we have ##f(x)=\sum c_k (z-z_1)^k = \sum d_k (z-z_2)^k## and as ##k \to \infty## the series equals f in both cases, but why would one choose a point...
Hello! I see that all theorems in complex analysis are talking about a function in a region of the complex plane. A region is defined as an open, connected set. If I am not wrong, the real line, based on this definition, is a region. I am a bit confused why there are so many properties of the...
Hello,I am a mechanical engineering student that loves mathematics and fluid mechanics. My school offers three different analysis courses and I’m not sure which to take. I took honors Fundamental of Mathematics, where we covered Abstract Linear Algebra, Set theory (along with rings and fields)...