Let (xn) be a bounded sequence in R. Show that there exist subsequences (xnk ) and(xmk ) of (xn) such thatlimk→∞ xnk = lim sup xn and limk→∞ xmk = lim inf xn.

To prove this, we will use the definitions of limit superior (lim sup) and limit inferior (lim inf) of a sequence.

1. Let's start with lim sup. By definition, lim sup of a sequence (xn) is the smallest number 'L' such that for any positive number ε, there exists a natural number N such that for all Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.