Let $M$ be a complete simply-connected Riemannian manifold with non-positive sectional curvature. Let $\gamma$ be a geodesic in $M$ and let $p$ be a point which does not lie on $\gamma$. Prove that the shortest distance between $\gamma$ and $p$ is realized by a unique geodesic which is perpendicular to $\gamma$
However, I do not think this solution is right. Because it claims at first that we can always achieve a minimum by quoting the result from problem 2, while problem 2 only applies to closed submanifolds. How do we actually show that there is always a critical point?