I need some help understanding this proof:

Prove: If a sequence converges, then every subsequence converges to the same limit.

Proof:

Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k \geq k$ for all $k$. This easy to prove by induction: in fact, $n_1 \geq 1$ and $n_k \geq k$ implies $n_{k+1} > n_k \geq k$ and hence $n_{k+1} \geq k+1$.

Let $\lim s_n = s$ and let $\epsilon > 0$. There exists $N$ so that $n>N$ implies $|s_n - s| < \epsilon$. Now $k > N \implies n_k > N \implies |s_{n_k} - s| < \epsilon$.

Therefore: $\lim_{k \to \infty} s_{n_k} = s$.

- What is the intuition that each subsequence will converge to the same limit
- I do not understand the induction that claims $n_k \geq k$