Let $\{f_k\}_{k \in \mathbb{N}} \subset C^1(\mathbb{R}^n; \mathbb{R})$ be a sequence of continuously differentiable functions s.t. $$\lim_{k \to + \infty} f_k(x) =0 \quad \forall \, x \in \mathbb{R}^n.$$
Is it true that $$\lim_{k \to + \infty} \inf_{x \in \mathbb{R}^n} |\nabla f_k(x)| =0$$ ? Here $\nabla f_k(x)$ is the gradient of $f_k$ at the point $x$ and $|\cdot|$ is the Euclidean norm.
It seems to me that the statement is true if $n=1$, but what about $n\ge 2$?