But how do we know that? Why can we just assert that all of the x's in K will belong to some B_{r}(p)? This seems to me to be the meat of the proof.
As wisvuze points out, of course if this is true then it will follow that there will be a finite number of such balls, so that we can then...
I came across this proof and have a question about the bolded portion:
Consider the following objection to the bolded: In order for \mathcal{G} to be an open cover of K its sets must contain all of the points of K. The sets of \mathcal{G} are B_r(p) for some fixed p, and so as r gets...
Thanks for your response lavinia. I'm still having trouble with the language here, so let me construct an example that I hope gets to the heart of my misunderstanding.
Suppose we're dealing with a 2-cell I which contains all points x = (x_1, x_2) such that a_1 \le x_1 \le b_1 and a_2 \le x_2...
I see that we have such intervals which are "taken" from the x-axis, y-axis, etc. But literally such intervals are now taken to the reals (as per definition of an interval being a set of real numbers), so that when united, do not form anything else but another interval.
Is Rudin just being...
First consider the following definitions from Baby Rudin:
Interval: A set of real numbers of the form [a,b] where for all x \in [a,b] we have a \le x \le b.
K-Cell: A set of k-dimensional vectors of the form x = (x_1, ...,x_k) where for each x_j we have a_j \le x_j \le b_j for each j from...
Thanks for your responses. I had some serious misconceptions about metric spaces that your examples help me clear up. Here are some conclusions (and one question) I've come up with:
(1) Whether a subset G is open is partially a function of its ambient space X. Pwsnafu's example shows this...
The following two definitions are taken directly from Rudin's Principles of Mathematical Analysis.
(1) OPEN SUBSET DEFINITION: If G is an open subset of some metric space X, then G \subset X and for any p \in G we can find some r_{p} > 0 such that the conditions d(p,q) < r_p, q \in X implies...
I see your point -- but I thought the goal was to think outside of the R2 metric. In Principles of Mathematical Analysis 2.24, Rudin makes a proof along these lines but NOT just for R2. So the question becomes: is there something that makes the proof necessarily true even in obscure metrics...
I'm still not getting it! Sure, N is defined as the smaller of N_{1} and N_{2}. So what? Why does that guarantee it will fit in X? Here is a visual aid to what I am sensing as a counter-example (see attached). In the picture we have N_{1} \subset A_{1} and N_{2} \subset A_{2} but clearly...
Suppose we have non-empty A_{1} and non-empty A_{2} which are both open. By "open" I mean all points of A_{1} and A_{2} are internal points. There is an argument -- which I have seen online and in textbooks -- that A_{1} \cap A_{2} = A is open (assuming A is non-empty) since:
1. For some x...
Well, here is my attempt at a proof that there exists an h with the two conditions Rudin lays out:
(1) 0 < h < 1
(2) 0 < h < [x – yn] / [n(y+1)n-1]
Let ⊥ = [x – yn] / [n(y+1)n-1]
First observe that ⊥ > 0 since it is really just the two positive quantities of (x-yn) and 1/[n(y+1)n-1] being...
Micromass and AlphaZero thank you for your replies. The ability to show the existence of elements within certain boundaries seems to be a pretty fundamental concept in Principles of Mathematical Analysis. I tried to answer both of your questions below to help my understanding.
Micromass...
Unique nth Roots in the Reals -- Rudin 1.21
In Principles of Mathematical Analysis 1.21, Rudin sets out to show that for every positive real x there exists a unique positive nth root y. The proof is rather long and I would like to zoom into the portion of it where it seems that Rudin takes too...