My professor has this proof in his notes and I would appreciate some feedback. It appears in the Mean Value Theorem section but I'm not sure if I need to incorporate that or not.

Let $\displaystyle I$ be an interval, suppose $\displaystyle f$ and $\displaystyle g$ are differentiable on $\displaystyle I$. If $\displaystyle f'(x)\leq g'(x)$ on $\displaystyle (a,b)$ and $\displaystyle c$ is in $\displaystyle I$, then prove that $\displaystyle f(x)-f(c)\leq g(x)-g(c)$ for all $\displaystyle x$ in $\displaystyle I$.

My attempt:

Let $\displaystyle f'(c)=(f(x)-f(c))/(x-c)$ and $\displaystyle g'(c)=(g(x)-g(c))/(x-c)$ as $\displaystyle x\rightarrow c$. Then, $\displaystyle f'(x)\leq g'(x)$ implies that $\displaystyle (f(x)-f(c))/(x-c)\leq g(x)-g(c))/(x-c)$. This implies that $\displaystyle f(x)-f(c)\leq g(x)-g(c)$ as $\displaystyle x\rightarrow c$ as desired.