Existence of a sub-sequence of non-converge sequence that $|x_{p_n}-L|>\epsilon$ > Let $\\{x_n\\}$ be a sequence that does not converge and let L be a real number. Prove that there exist $\epsilon >0$ and a sub-sequence $\\{x_{p_n}\\}$ of $\\{x_n\\}$ such that $|x_{p_n}-L|>\epsilon$ for all n. I don't have any idea on how to prove this. Any advice and suggestions will be greatly appreciated.

Since $(x_n)$ does not converge to $L$ there is $\epsilon >0$ such that

$|x_n-L|> \epsilon$ for infinitely many n.

Now, its your turn.