Let $D_f (x)$ be the set of descent directions of $f$ at $x \in \mathbb R^n$.
Prove that $$ d \in D_f (x) \Rightarrow \langle \nabla f(x), d\rangle $$ and
if $\langle \nabla f(x), d\rangle < 0$ then $d \in D_f (x)$
I have an idea of proving this with a linear Taylor approximation, but it doesn't seem to formal.
If $d$ is a descent direction then for $\varepsilon$ small enough, $$\frac{f(x+\varepsilon d) - f(x)}{\varepsilon} \leq 0.$$ This implies by taking the limit $\varepsilon \rightarrow 0$ that $$\langle \nabla f(x) ,d \rangle \leq 0.$$
Alternatively, if $\langle \nabla f(x) ,d \rangle < 0$ then for some $\varepsilon > 0$, by continuity/definition of the limit, we must also have $$\frac{f(x+\varepsilon d) - f(x)}{\varepsilon} \leq 0.$$ This is no longer necessarily true if we only have $\langle \nabla f(x) ,d \rangle \leq 0$.