currently I am reviewing the diagonal trick regarding real analysis, i.e. one can prove that for example if $V=\{F: \mathbb{R} \to \mathbb R : \text{ right cont. , monotone increasing and bounded} \}$ then there exists a subsequence, which I donote by $\Lambda$ such that $$ F_n(x) \to F(x) \; \left(n \to \infty , n \in \Lambda \right). $$
In the start of the proof one uses Bolzano-Weierstrass to iteratively construct subsequences such that for an enumaration of $\mathbb Q=\{q_1, q_2, \dots \}$ one has that for all $l \in \mathbb N$ there exists a subsequence $\Lambda_l \subseteq \Lambda_{l-1} \dots \subseteq \Lambda_2 \subseteq \Lambda_1 \subseteq \mathbb N$ such that:
$$ F_n(q_l) \to F(q_l) \; \left(n\to \infty \; n \in \Lambda_l \right) $$
At this point I one already constructed a converging sequence for a $\textbf {fixed but finite}$ number of rational points $\{q_1 , \dots , q_l\}$. In order to $\textbf{push it to all rational points}$ one considers the sequence $n \mapsto \Lambda^{(n)}_n$ (n-th element in the n-th subsequence). This is a sufficient subsequence.
Now here comes my question: Can one also choose another subsequence in the end i.e. is $n \mapsto \Lambda^{(n)}_{n+1}$ or $\Lambda^{(n)}_{e^n}$ also sufficient? They should be as long one iterations over all subsequences.