Definition: A subsequence $(a_{n_k})$ of $(a_n)$ is convergent if given any $\epsilon >0$, there is an $N$ such that $\forall k\geq N \implies\vert a_{n_k} - \ell\vert < \epsilon$
Why do we want $k \geq N$ and not $n_{k} \geq N$?, Don't they both refer on the same "last" term of your subsequence?
Or am I getting confused that $n$ is actually 'fixed' and it is only used to refer back to our original sequence $(a_n)$? And what is referring to the terms of my subsequence are actually the $k$s?
The subsequence, written out, is $\langle a_{n_0},a_{n_1},a_{n_2},a_{n_3},\ldots\rangle$; the indices are $n_0,n_1,n_2$, and so on. In other words, it’s the $k$ in $a_{n_k}$ that tells you which term of the subsequence you have; the $n$ is a fixed symbol that does not actually represent an integer by itself.