Proof of convergent Cauchy Sequence, if sime subsequence converges to p ![enter image description here]( The idea is clear. But can you explain me please in more detail the procedure of chosing $N$ for our subsequence. Why $N$ must be greater or equall $N_1$? And why $n_{N+1} >N+1 >N_1$?

The rank for which your sequence has consecutive terms very close is $N_1$, that means after $N_1$ (ie: for $n\geq N_1$), all terms are close to each other.

Also you know that after $N$, all the terms of your subsequence and $p$ are close.

If you want both properties to be true, you must choose $n$ greater than $N_1$ and $N$.

For your last question, remember that an extraction $\phi$ is such that $\phi(n)>n$. In this redaction, $n_k > k$. In this redaction, the author made the choice to directly take $N$ greater than $N_1$, since it's true for all rank greater than $N$, you must always consider it greater than $N$.