A proposition in my textbook essentially says $a_n \rightarrow a$ iff $a_{n_k} \rightarrow a$ for all subsequences $(a_{n_k})$ of $a_n$, but the proof is confusing me.
The proof is as follows (condensed so it isn't too long):$(a_n) \rightarrow a$ so for $ε>0$ there exists some $N>0$ st $|a_n-a| < ε$. (1)
We want to show that there exists some $N_1$ st $\forall k>N_1$$|a_{n_k}-a| < ε$ (2)
Notice that since each $n_i \in \mathbb{N}$ and $n_1<n_2<n_3...$, that we have $n_k \geq k$. (3)
Therefore by letting $N_1=N$, for any $k>N_1$ we then know that $n_k>N$ (4)
, so since $|a_n-a| < ε$, we have $|a_{n_k}-a| <ε$ .
(the backwards direction is left as an exercise)
My confusion is on line 2. By the definition of sequence convergence, wouldn't we be looking for an $N_1$ st for all $n_k>N_1$, $|a_{n_k}-a| <ε$? Why do they say for $k>N_1$?
Thanks for the help!