Hopf-Rinow theorem from do Carmo's book, proof problem.

Question

I'm reading do Carmo's book Riemannian geometry and I have a problem. At the proof of Hopf-Rinow theorem, more precisely at c.)=>d.) ($M$ complete metric space => $M$ geodezically complete space). I don't Understand the last step of the proof.

Namely the fact that $\text{exp}_{\gamma(s_n)}(B_\delta(0))\supset W.$ And why $g$ extends $\gamma$ beyond $s_0?$