I asked a while ago a similar question about this topic. But doing some exercises and using this stuff, I still get stuck. So I have a new question about this topic. (Here is the link for the previous question: Cantor diagonalization method for subsequences).
Now my new question: Suppose I have a sequence $(x_n)$, a set $K$ and a function $f$ and we define for all $l \in \mathbb{N}$ the set $M_l:= K\cap \{x|f(x)\le l\}$ with $f< \infty$ on $K$. I've proved that for a fixed $l$ there's a subsequence $(x_{n_k})$ converging on $M_l$, denote this by $(x^l_{n_k})$. First it's clear that $M_l \subset M_{l+1}$ and I want to show that there's a subsequence which converges on $\cup_{l\ge1} M_l = K$.
So I have different subsequences $(x^1_{n_k}),\dots,(x^p_{n_k}),\dots$ and define the diagonal sequence as ${x^{\phi(l)}_{n_{\phi(l)}}}$ where $\phi(l)$ is the $l$-th element of $n_k$ (just the diagonal sequence). Is it enough to say, since $(M_l)$ is increasing the sequence converges on $K=\cup_{l\ge1} M_l$?
I'm sorry, but I'm just confues about picking the right sequence and to prove that it is the right sequence using Cantor. Thank you for your help