Let function $f: \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable, lower-bounded and convex. Let $\nabla f$ be the gradient of $f$, i.e., $\nabla f = \begin{bmatrix} \frac{\partial f}{\partial x_1} \dots \frac{\partial f}{\partial x_n} \end{bmatrix}$. If $x^1,x^2,\dots$ is a sequence in $\mathbb{R}^n$ such that
$$\lim_{n\to \infty} f(x^i) = \inf_{x \in \mathbb{R}^n} f(x)$$
then is the following true?
$$\lim_{n\to \infty} \left\| \nabla f(x^i) \right\| = 0$$
where $\|\cdot\|$ is the usual Euclidean norm. If this is not true, what additional assumptions on $f$ can be added so that this is true?
Thank you.
Take $n=2$ and consider $f(t_1,t_2) = \max(e^{-t_1}, |t_2|)$. Being the maximum of two convex functions, this is convex. The points $x^i = (i, 2 e^{-i})$ have $f(x^i) = 2 e^{-i} \to 0 = \inf f$, but $|\nabla f(x^i)| = 1$.
This example is not continuously differentiable, but can be smoothed.