This is intuitively clear, but I cannot solve this homework problem:
1) Let $(M,g)$ be a complete Riemannian manifold, let $c:[0,1]\to M$ be a continuous curve in $M$ such that $c(0)=p, c(1)=q$. Then prove that in the fixed-end-point homotopy class of $c$, there is a geodesic $\gamma$, i.e. there exists a geodesic $\gamma$ so that $\gamma$ is homotopic to $c$ with homotopy keeping the end points $p,q$ fixed.
2) My question: Assuming the above is true, is that geodesic $\gamma$ in the answer necessarily minimizing as well? I feel it should be.
I was thinking of using Hopf-Rinow theorem stating that geodesically complete is the same as metrically complete and starting with the contrary. But I got stuck.