Let $E$ be a subset of $\textbf{R}^{n}$, $f:E\to\textbf{R}^{m}$ be a function, $x_{0}$ be an interior point of $E$, and let $v$ be a vector in $\textbf{R}^{n}$. If $f$ is differentiable at $x_{0}$, then $f$ is also differentiable in the direction $v$ at $x_{0}$, and \begin{align*} D_{v}f(x_{0}) = f'(x_{0})v \end{align*}
My solution
According to the definition of differentiability, we have \begin{align*} \lim_{x\to x_{0}}\frac{\|f(x) - (f(x_{0}) + f'(x_{0})(x-x_{0}))\|}{\|x-x_{0}\|} = 0 \end{align*}
If we make the substitution $x = x_{0} + tv$, we have that \begin{align*} \lim_{t\to 0^{+}}\frac{1}{\|v\|}\left\|\frac{f(x_{0} + tv) - f(x_{0})}{t} - f'(x_{0})v{}\right\|= 0 \end{align*} which implies that $D_{v}f(x_{0}) = f'(x_{0})v$, and we are done.
Could someone please check the wording of my proof. Is it correct? Is there any theoretical flaw?