In this week, we finished section 10 on monotone sequence and Cauchy sequence, and also touches a bit on constructing subsequences. It is important to understand the statements of the propositions/theorems that we talked about in class, and then try to prove them yourselves, then compare with notes and textbook.

1. Let $(s_n)$ be a bounded sequence.

• (a) Show that $\limsup s_n \geq \liminf s_n$.
• (b) Show that $\limsup s_n = \inf_{N} \sup_{n \geq N} s_n$.