I am working on a proof involving uniform continuity on functions with bounded domains. In particular $f:(a,b)\rightarrow\mathbb{R}$. I have an idea but it requires me to establish the following property.
Let $\{x_n\}$ be a bounded divergent sequence. Prove that there exists a convergent subsequence of $\{x_n\}$ call it $\{x_{n_k}\}$ such that $lim_{k\rightarrow\infty}[x_k-x_{n_k}]=0$
Can anybody find a proof or provide a counterexample?