comparing compactness theorems: bolzano-weierstrass and arzela-ascoli

Question

Rudin says In [BW] we saw that every bounded sequence of complex numbers contains a convergent subsequence, and the question arises whether something similar is true for sequences of functions. I'm trying to follow his statement and understand the role of equicontinuity in Rudin's Arzela-Ascoli theorem in the framework of Bolzano-Weierstrass.

For reference, Rudin's AA theorem states:

If $K$ is compact, if $f_n \in \mathcal{C}(K)$ for $n=1,2,\ldots$, and if $\{f_n\}$ is pointwise bounded and equicontinuous on $K$, then

1. $\{f_n\}$ is uniformly bounded on $K$
2. $\{f_n\}$ contains a uniformly convergent subsequence

and his BW states:

1. If $\{p_n\}$ is a sequence in a compact metric space $X$, then some subsequence of $\{p_n\}$ converges to a point of $X$.
2. Every bounded sequence in $\mathbb{R}^k$ contains a convergent subsequence.

For BW, I have a simple mental picture of two points in a closed/bounded interval of $\mathbb{R}$, a sequence that alternates between the two points, and a convergent subsequence constructed by taking a sequence of one of the points.

I'm looking for another simple view of AA similar to this so that I can develop some more tangible idea of what equicontinuity provides. Does anyone have a nice, simple interpretation?

Answer 1

(copy/pasted from comments)

Your view of BW should be more general. The point is that if you have any infinite sequence in this interval, there is some subsequence which converges. The points can be spread out quite sporadically. Somehow the compactness says that there is 'finite space' and you are trying to put infinitely many points in this space. So somehow they accumulate. On the other hand, you can think of $C(K)$ as a metric space itself, with metric induced by say, the sup norm. This space is not itself compact, but if it is bounded and equicontinuous, then it becomes (relatively) compact. The general form of BW says that a sequence of points in a compact metric space has a subsequence, so really Arzela-Ascoli gives conditions on when $C(X)$ is a compact metric space - when the functions are bounded and equicontinuous.