I'm having trouble with this one, I don't know what is the right path to solve it.
$f$ is a differentiable function in $(a,b)$, and let $x_0 \in (a,b)$. prove that there exists a sequence $(x_n)$ so that: $x_n \neq x_0$ for every $n$, $\lim_{n->\infty} x_n=x_0$ and $\lim_{n->\infty} f'(x_n) = f'(x_0)$
A hint I have: for every natural $n$, regard $[x_0,x_0+1/n]$ and use Lagrange's mean value theorem.
I tried to use the property and definition of a derivative to prove that $\lim_{n->\infty} f'(x_n) = f'(x_0)$, but couldn't find a way to conclude that $\lim_{n->\infty} x_n=x_0$. I could assume that, but I don't know how to conclude that or how to prove it.
Hoping you could help me. Thank you very much!