I have been trying to solve the following exercise from Abbott's "Understanding Analysis".
I understand that $(a)$ comes from an application of the Bolzano Weierstrass Theorem as we assume that $f_n (x)$ is bounded for all $n$ and $x$. But after that I do not understand how to proceed. Moreover the notation confuses me. If we have taken $f_{1,k}$ to be the convergent subsequence of functions evaluated at $x_1$ then what does $f_{1,k} (x_2) $ stand for?
Thank you.
The lower double index confused me in a first read but then it becomes way clearer than the "natural" alternative )index of index of index...): the book begins with $\;f_{n_1}\;$, i.e. one single index with one subindex, so I think the intention is:
Prove there exists a subsequence of $\;\{n_k\}\subset\{n\}\;$ s.t. that $\;\{f_{n_k}(x_1)\}:=\left\{f_{1,k}(x_1)\right\}\;$ converges. This is straighforward B-W and we get a converging subsequence as defined by Abbot just above (b) in his book.
Since the sequence $\;\left\{f_{1,k}(x_2)\right\}\;$ is bounded, there exists a subsequence $\;\left\{{2,k}\right\}\subset\left\{{1,k}\right\}\;$ s.t. $\;\left\{f_{2,k}(x_2)\right\}\;$ that converges.
Continues as above: there exists a subsequence $\;\left\{f_{3,k}(x_3)\right\}\;$ that converges and etc.