Last time, we ended at discussion of two equivalent definitions of subsequential limit. I hope the Cantor's diagonalization trick was fun. Today, we prove the following results

• If $s_n$ converge to $s$, then every subsequence of $s_n$ converges to $s$.
• Every sequence has a monotone subsequence.
  • If there are infinitely many $s_n$, that is 'larger than its tails' ($s_n \geq s_k$ for all $k \geq n$), then we can take the subsequences of such $s_n$, it is monotone decreasing.
  • Otherwise, you throw away an initial chunk that contains such $s_n$, then you can always build an increasing sequence (nothing can stop you)
• Every bounded sequence has convergent subsequence. (just take a monotone sequence, then it will be convergent)
• $\limsup$ and $\liminf$ can be realized as subseq limits.
• Let $(s_n)$ be a seq, and $S$ denote the set of all subseq limits. Then, $S$ is non-empty. $\sup S = \limsup s_n$ and $\inf S = \liminf s_n$.