## Existence of Directional Derivative for an interior point

Suppose that $f: A \subset \mathbb {R}^n \rightarrow \mathbb {R}^m$ is differentiable at $x_0 \in Int(A)$. Prove that for every $v \in \mathbb {R}^n$, $\lim _{t \rightarrow 0} \frac {f(x_0+tv)-f(x_0)}{t}$ exists and is equal to $Df_{x_0}(v)$

My work:

Let $x_0 \in int(A)$, then there exists an open set $U$ such that $x_0 \in U \subset \mathbb {R}^n$. Let $v \in \mathbb {R}^n$

Find $\epsilon > 0$ such that $||x_0- \epsilon || \subset U$

Pick $t \in \mathbb {R}$ such that $|t| ||v|| < \epsilon$

Then there exists $z_1,z_2,...,z_n \in \mathbb {R}^n$ such that $||x-z_k||<||tv||=|t|||v|| \ \ \ \ \ \forall k \in \{ 1,2,...,n \}$ and $f(x+tv)-f(x)= \sum ^N _{k=1}tv_k \frac { \partial f }{ \partial x_k } (z_k)$

Am I doing this right so far? Thanks.