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. enter image description here

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

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    $\begingroup$ This is the definition of normal neighborhood (see Proposition 3.6) and a totally normal neighborhood (see Theorem 3.7). $\endgroup$ – M. Dus Aug 14 '18 at 9:01
  • $\begingroup$ Yeah, you are right! Thanks! $\endgroup$ – Hurjui Ionut Aug 14 '18 at 10:50

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