If xn converges to zero then every subsequence of xn converges to 0

I posted a proof that we did in class and I wrote *** to the place that I don't understand. Can someone explain me that part?

He suddenly said there exists k such that nk>N . According to what?

Is the logic that we say, ok let's take an increasing sequence nk, that corresponds to the elements of n.

And therefore there exists k such that nk>N

So for all k>K we have nk>nK because it's an increasing sequence and since the sequence nk that we created correspondes to the elements of n we will get |xnk|<epsilon

Is my logic correct?

Note:My analysis exam is coming up so I might ask the questions more than usual, about the things that I don't understand.Therefore I ask for your patience. I will always write my thinking process so if I have something wrong in my thinking I hope you can fix me. Thank you!