Subseqeunce convergence definition

Subseqeunce convergence definition

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?