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 $\limsup s_n = \inf_{N} \sup_{n \geq N} s_n$.

2. Let $(a_n), (b_n)$ be two bounded sequences, show that $$\limsup (a_n + b_n) \leq \limsup(a_n) + \limsup(b_n)$$ and give an example where the inequality is strict.

3. 10.6

4. 10.7

5. 10.8

6. 10.11

7. Let $S$ be the subset of $(0,1)$ where $x \in S$ if and only if $x$ has a finite decimal expression $0.a_1 a_2 \cdots a_n$ for some $n$, and the last digit $a_n=3$. Show that for any $t \in (0,1)$, there is a sequence $s_n$ in S that converges to $t$.