Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a $C^2$ convex function which is strictly positive. If $x_n$ is a sequence of points such that $f(x_n)\rightarrow 0$, show that (or give a counterexample) the gradient $\nabla f(x_n)$ also tends to zero.

A counterexample is $$f=\sqrt{y^2+e^{x}}.$$ You can verify by computing the second derivatives that this is convex. As a sequence $x_n$ you can take $(n,1/n)$. Then $f(x_n)\to 0$ but the derivative with respect to $y$ tends to 1. Thus the gradient does not tend to 0. 

