two examples in analysis I want to ask for two examples in the following cases:
1) Given a bounded sequence $\{a_n\}$, $$\lim_{n\to \infty}{(a_{n+1}-a_n)}=0$$ but $\{a_n\}$ diverges.
2) A function defined on real-line $f(x)$'s Taylor series converges at a point $x_0$ but does not equal to $f(x_0)$.
Thanks for your help.
Edit
in 2), I was thinking of the Taylor series of the function $f$ at the point $x_0$.

2) It is classical:
$$f(x)=\begin{cases} \exp (-\tfrac{1}{x}) &\text{, if } x>0 \\ 0 &\text{, if } x\leq 0 \end{cases}$$
and $x_0=0$.
