Reference request for Heine-Borel theorem

Question

I would like to know a nice reference for the Heine-Borel theorem. In a text, I have the compactness argument for the following two sets. The reference should be able to cover these two cases.

1. The closed ball $\mathcal{G}=\{g:D(f,g)\leq \epsilon\}$, where $f$ is a known density function defined on the real numbers, $D$ is a convex distance and $\epsilon$ is a small number. This closed ball is not compact because it is not finite dimensional. I just quantize all density functions to $N<\infty$ discrete levels on the domain of density functions. That is, I divide the real numbers into $N$ levels and integrate the density in that interval. E.g. let $g$ be the standard Gaussian density $N=3$, so the intervals are (for example) $(-\infty -1]$, $(-1,1)$ and $[1,\infty)$. if one integrates $g$ in these intervals, then we get a discrete approximation of the Gaussian density. If we apply the same story to all $g\in\mathcal{G}$, then $\mathcal{G}$ is finite dimensional, closed and bounded, hence according to Heine-Borel theorem it is compact.

2. The set of all functions on the real numbers and in the range $[0,1]$. Let this set be $\Delta$. $\Delta$ is compact with respect to product topology according to Tychonoff's theorem. It is however not sequentially compact. I would like to quantize all $\delta\in\Delta$ in the same way as above. All functions $\delta$ will be discrete having values in $[0,1]$. So $\Delta$ will be finite dimensional, bounded and closed. Again by the application of Heine-Borel theorem, I want to claim compactness.

What are your recommendations? It seems this is an old theorem and well known but not well known out of mathematics and I cannot give wikipedia as reliable source in a paper.

Answer 1

How about Theorem 2.41 in Rudin's book (often referred as "baby Rudin")?

Answer 2

Heine-Borel's (or Borel-Lebesgue's in France) asserts in a Euclidean space, a subset is compact if and any if it is closed and bounded.

This theoem is false for infinite dimensional spaces.

Counter-example:

Consider the vector space $\mathbf R[x]$, and for any polynomial $p(x)=a_0+a_1x+\dots+a_nx^n$, set $\;\lVert p\rVert=\max(\lvert a_1\rvert, \dots,\lvert a_n\rvert)$. In this vector space, the sequence $(x^n)$ has no convergent subsequence, since $\lVert x^m -x^n\rVert=1$ for any $m\ne n$.