What shown below is a reference from "Topology" by James Munkres

[![enter image description here](https://i.stack.imgur.com/7wACB.png)](https://i.stack.imgur.com/7wACB.png)

So I don't understand why $\overline{S}=S\cup\{(0,t)|t\in[-1,1]\}$ and why it is possible to choose $u\in(0,x(\frac{1}n))$ such that $\text{sin}(\frac{1}u)=(-1)^n$ and then why by the intermediate value theorem it follows that there exist $t\_n\in(0,\frac{1}n)$ such that $x(t\_n)=u$ and so why $t\rightarrow 0$ and $y(t\_n)=(-1)^n$. So could someone help me, please?