## Bolzano–Weierstrass for bounded functions

Need help understanding this proof please

Theorem
If $[f_n]$ is a pointwise bounded sequence of complex functions on a countable set $E$, then $[f_n]$ has a subsequence $[f_{n_k}]$ such that $[f_{n_k}(x)]$ converges $\forall{x\in{E}}$

Proof
Let $[x_i], i=1,2,3...$, be the points of $E$, arranged in a sequence. Since $[f_n(x_1)]$ is bounded, $\exists$ a subsequence, which we shall denote by $[f_{1,k}]$ such that $[f_{1,k}(x_1)]$ converges as $k\rightarrow\infty$

Let us now consider sequences $S_1,S_2,S_3,...$, which we represent by the array

$S_1$: $f_{1,1}$ $f_{1,2}$ $f_{1,3}$ $f_{1,4}$ ....
$S_2$: $f_{2,1}$ $f_{2,2}$ $f_{2,3}$ $f_{2,4}$ ....
$S_3$: $f_{3,1}$ $f_{3,2}$ $f_{3,3}$ $f_{3,4}$ ....
.................................

and which have the following properties:
(a) $S_n$ is a subsequence of $S_{n-1}$ for $n=2,3,4,...$

(b) $[f_{n,k}(x_n)]$ converges, as $k\rightarrow\infty$ (the boundedness of $[f_n(x_n)]$ makes it possible to choose $S_n$ in this way)

(c) The order in which the functions appear is the same in each sequence; i.e, if one function precedes another in $S_1$, they are in the same relation in every $S_n$, until one or the other is deleted. Hence, when going from one row in the above array to the next below, functions may move to the left but never to the right.

We now go down the diagonal of the array; i.e, we consider the sequence

$S$: $f_{1,1}$ $f_{2,2}$ $f_{3,3}$ $f_{4,4}$

By (c), the sequence $S$ (except possibly its first $n-1$ terms) is a subsequence of $S_n$, for $n=1,2,3,..$ Hece (b) implies that $[f_{n,n}(x_i)]$ converges,, as $n\rightarrow\infty$, $\forall{x_i}\in{E}$

QED

Questions
1) For $S_m$: $f_{m,1}$ $f_{m,2}$ $f_{m,3}$ $f_{m,4}$ ....
Are the terms of this sequence, S_m, the terms of the sequence $[f_{m,k}]$, the convergent subsequence of the sequence $[f_n(x_m)]$?

If question 1 is true

2) Regards to property (a), is this a property intrinsic to an array of sequences constructed in this manner; i.e an array of sequences made of the terms of $[f_{n,k}]$, or must the array be specially constructed to have this property?
If so, how do I guarantee this can always be done?

3) Would't property (a) imply $f_{n,j}=f_{n-1,p}$ for all $j=1,2,3...$ and some $p\in{\mathbb{N}}$?