To prove (partially)
If the set $\{n\in \mathbb{N}:|s_{n}-t|<\epsilon\}$ is infinite for all $\epsilon > 0$, then there is a subsequence of $s_{n}$ converging to $t$.
the author assumed $\{n\in \mathbb{N}:t-\epsilon <s_{n}<t\}$ is infinite for all $\epsilon >0$, and constructed a subsequence $s_{n_{k}}$ such that
- $t-1<s_{n_{1}}<t\,$,
- $\max\{s_{n_{k-1}},\,t-\frac{1}{k}\}\le s_{n_{k}}<t\quad\text{for}\quad k\ge 2$.
However, is it really necessary that $s_{n_{k-1}}\le s_{n_{k}}<t\,$? Aren't $t-\frac{1}{k}\le s_{n_{k}}<t$ enough, by which we can apply Squeeze Lemma?