Let $E$ a countable metric space then $E= \{x_n\}$, let $\{f_n\}$ a sequence of functions $f_n: E \to \mathbb{R}$, such that for every $x \in E$ exists $M_x$ s.t. $|f_n(x)|< M_x$ for every $n \in \mathbb{N}$. Then exists a subsequence $\{f_{n_k}\}$ pointwise convergent.
I can't solve this exercise coud you suggest some references? Thank you!
This is immediate from 'Cantor's diagonal procedure'. There is a subsequence $(n^{(1)}_k)$ such that $f_{n^{(1)}_k}(x_1)$ converges. [ Because any bounded sequence in $\mathbb R$ has a convergent subsequence]. Now look at $f_{n^{(1)}_k}(x_2)$. There is a subsequence $(n^{(2)}_k)$ of $(n^{(1)}_k)$ such that $f_{n^{(2)}_k}(x_2)$ converges. Note that $f_{n^{(2)}_k}(x_1)$ also converges. Repeat this process. You get subsequences $(n^{(j)}_k), j=1,2,...$. Define $n_j$ to be $n^{(j)}_j$. This subseqeunce has the required property.