Rudin PMA p.157
I'm trying to prove;
"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 $\{f_{n_k}\}$ pointwise convergent on $E$"
It's clear that Rudin made dependent choice.
I'm trying to prove this withouc AC and want to know if there is a way to choose a subsequence of a sequence. That is when $\{n\}$ is a given sequence, then i want to choose a subsequence $\{n_k\}$, then again choose a subsequence of $\{n_{k_j}\}$ and again countable times.
Is it possible?
Or if one can prove this with a different argument, Or if it is unprovable, please let me know..
Thank you in advance