Gradient of convex functions

Question

Let $f$ be a convex and differentiable function in $\mathbb{R}^n$. Then if we have a minimizing sequence $\{x_k\}$ (i.e. $f(x_{k+1})\le f(x_k)$), and if we also know that $$\liminf_{k\to\infty}\|\nabla f(x_k)\|=0,$$ will we get $$\lim_{k\to\infty}\|\nabla f(x_k)\|=0~?$$ If not, will it be true if we add L-smoothness or strong convexity?