Are following statements true or false ?
If function $f$ is differentiable at $x_0$, then the sequence n.$( f(x_0+(1/n)) - f(x_0))_{n\in\mathbb{N}}$ is convergent.
I am really not sure, but I think it is true, but just because it seems to me so and I can't imagine any proof of it, would you help me ? Maybe it has something with Heine's sentence (a transition from function to sequence) but I see nothing in it. Thanks for your help. :)
Dfferentiability at $x_0$ implies continuity at $x_0$. In particular, $$\lim_n f\left(x_0+\frac{1}{n}\right) - f(x_0) = \lim_n \frac{f\left(x_0+\frac{1}{n}\right) - f(x_0)}{\frac{1}{n}} \cdot \frac{1}{n}.$$ You can evaluate each limit separately (limit of product is product of limits) to get $\lim_n \cdot = f^\prime(x_0) \cdot 0 =0$. So, yes the sequence converges (to zero).
Edit: It seems that I missed the $n$ in front. In that case, $$\lim_n n\cdot \left(f\left(x_0+\frac{1}{n}\right) - f(x_0)\right) = \lim_n \frac{f\left(x_0+\frac{1}{n}\right) - f(x_0)}{\frac{1}{n}} = f^\prime(x_0).$$ Now that I've addressed the right question, does his make sense?