Problem 13.17 (Every connected smooth manifold admits a complete Riemannian metric)

Let $M$ be a connected noncompact smooth manifold and let $g$ be a Riemannian metric on $M$. Prove that there exists a positive function $h\in$ $C^{\infty }(M)$ such that the Riemannian metric $\tilde{g}=\mathit{hg}$ is complete. Use this to prove that every connected smooth manifold admits a complete Riemannian metric. [Hint: let $f:M\to ℝ$ be an exhaustion function, and show that $h$ can be chosen so that $f$ is bounded on $\tilde{g}$-bounded sets.]