Proof of Hopf Rinow theorem

Question

I can't understand the red line , it just prove converge to the closure.

Answer 1

If $a_n$ is Cauchy, and there is a subsequence $a_{n_j}$ that converges to $a$, then $a_n$ converges to $a$. The reason is that for any $\epsilon$, there is a $M$ so that all $a_n$ are within epsilon of eachother, and there is some $n' \geq M$ so that $a_n'$ (in the subsequence) is within epsilon of $a$. Then use the triangle inequality.

Answer 2

When compact are closed and bounded, Cauchy sequence are bounded, and thus, using Bolzano Weierstrass, there is a subsequence $(x_{n_k})$ that converge. Let denote $\ell$ it's limit. Then, using Cauchy property (and the fact that $n_k\geq k$), you get that $$d_g(x_n,\ell)\leq d_{g}(x_k,x_{n_k})+d_g(x_{n_k},\ell)<\varepsilon$$ when $k\geq N$ for a certain $N$, what prove the claim.