Gradient points in the direction of steepest ascent

Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be differentiable at $a \in U$, and suppose that $\nabla f(a) \ne 0$. Then for each unit vector $u \in \mathbb{R}^n$, the directional derivative satisfies $-\|\nabla f(a)\| \le D_u f(a) \le \|\nabla f(a)\|$. The upper bound is attained if and only if $u = \nabla f(a) / \|\nabla f(a)\|$, and the lower bound if and only if $u = -\nabla f(a) / \|\nabla f(a)\|$.