A problem in Riemannian geometry (related geodesic)

A geodesic r:[0,inf) ->M in a Riemannian manifold M is called a ray
starting from r(0) if it
minimizes the distance between r(0) and r(s), for any s in (0,inf).
Assume that M is complete, non compact, and let p in M. Show that M
contains a ray starting from p.


**I have a plasible candidate for such ray, that is,
since M is non- compact, by Hopf-Rinow thm, M should be unbounded.
Therefore, there exist sequence {q_n} such that d(p,q_n)->infinity as
n->inf.
Furthermore, M is complete, there is minimal normalized geodesic
r_{v_n}=exp_p (t*v_n) between p and q_n.
Since |v_n|=1, there exist accumulation vector v of {v_n} in S^1.
The candidate I guess for ray is exp_p (t*v).

But I cannot prove it.
Can you help me?
.

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