Let $u_n$ be a bounded sequence such as $\displaystyle\lim_{x\to +\infty}\,u_{n+1}-u_n=0$. Can we say that $u_n$ is convergent just from these 2 conditions?.
My take on this was to prove that $u_n$ is a Cauchy sequence by proving that $\displaystyle\lim_{n\to +\infty}\,u_{n+1}-u_n=0 \implies\displaystyle\lim_{n\to +\infty}\,u_{n+p}-u_n=0$ for all $p\in\mathbb{N}$, using the definition of the limit of a sequence.
The reasoning goes as follows:
$\displaystyle\lim_{n\to +\infty}\,u_{n+1}-u_n=0\implies (\forall\epsilon>0)(\exists N\in\mathbb{N})(\forall k\geq N)\;\left|u_{k+1}-u_k\right|\leq\epsilon$. By summing from n to n+p we have $(\forall\epsilon>0)(\exists N\in\mathbb{N})(\forall k\geq N)\;\left|u_{n+p}-u_n\right|\leq p\epsilon$.
But here is where my teacher said that I have made a mistake, since now our new $\epsilon'=p\epsilon$ depends on p, which wouldn't imply that $\displaystyle\,u_{n+p}-u_n$ converges to 0.
Geometrically, I thought that if $\displaystyle\lim_{n\to +\infty}\,u_{n+1}-u_n=0$ then $u_n$ will station itself as n goes to infinity, and since its bounded then it will station itself on a finite value.
