Diagonal subsequence converges at all points under these conditions

Question

I do not understand why the diagonal subsequence $g_n = f_{n,n}$ converges at any point $w_j$. for example, because $g_{n} (w_{20})$ it converges? From what I understand we would have $\{ f_{1,1} (w_{20}), f_{2,2} (w_{20}) , f_{3,3} (w_{20}), ... \}$. Why does it converge? If I'm wrong, someone can explain?

Answer 1

Except for the first 19 terms, the sequence $(g_n)$ is a subsequence of $(f_{n,20})$, which converges at $w_{20}$.