Question in do Carmo's book Riemannian geometry section 7 I have a question. Please help me.

Assume that $M$ is complete and noncompact, and let $p$ belong to $M$. 
  Show that $M$ contains a ray starting from $p$.

$M$ is a riemannian manifold. It is geodesically and Cauchy sequences complete too. A ray is a geodesic curve that its domain is $[0,\infty)$ and it minimizes the distance between start point to each other points of curve. 

Take a sequence of points $(x_i)$ in the manifold whose distance from $p$ tends to infinity, and connect each of them to $p$ by a minimizing geodesic $\gamma_i(s)$.  Choose a convergent subsequence $\gamma'_{i_k}(0)$ at $p$.  Then the limit of the sequence is the desired direction of a ray.
