Let $F : \mathbb{R}^n \to \mathbb{R}^m$. If there exists an $m \times n$ matrix $A$ and $\rho : \mathbb{R}^n \to \mathbb{R}^m$ such that $\forall h \in \mathbb{R}^n$, $F(a+h) = F(a) + Ah + ||h||\rho(h)$, where $\rho(h) \to 0$ as $h \to 0$, then $F$ is differentiable at a.
My attempt: Rearranging yields $\dfrac{F(a+h) - F(a) - Ah}{||h||} = \rho(h)$, and $\rho(h) \to 0$ means the left hand goes to $0$, and that's the definition of differentiability (if $A$ is the Jacobian).
This almost seems too easy but I feel like I'm missing many details: could someone help me out in filling the missing assumptions?
The only think is that you also need to prove that $h \mapsto Ah$ is continuous. Which is the case as a linear map defined on a linear space of finite dimension is always continuous.