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.