convergence of a subsequence of function for a given rational in a closed interval

Enumerate rationals $\{q_j\}_j \subset [a,b]$. Start with $q_1$. Since $|f_n(q_1)| \le 1$ you can find subsequence $\{n_k^{(1)}\}$ such that $f_{n_k^{(1)}}(q_1) \to v_1$, where $v_1 \in [-1,1]$ is some number (we used the fact that from bounded sequence in $\mathbb R^d$ we can extract converging subsequence (Bolzano - Weierstrass)). Now, from $\{n_k^{(1)}\}$ by the same reasoning we can extract subsequence $\{n_k^{(2)}\}$ such that $f_{n_k^{(2)}}(q_2) \to v_2 \in [-1,1]$. Repeat the same for every rational $q_j$ and finally take $n_k = n_k^{(k)}$. Since for any $j \in \mathbb N_+$ it's a subsequence (starting from index $k \ge j$) of $n_k^{(j)}$ we get $f_{n_k}(q_j) \to v_j$ for any $j \in \mathbb N_+$.