1. ## closure problem

Let (X,p) be a metric space and let S Í X be totally bounded. Argue that the closure of S is totally bounded.

2. Originally Posted by jburks100
Let (X,p) be a metric space and let S IÍ X be totally bounded. Argue that the closure of S is totally bounded.
Suppose that $\overline{S}$ is not totally bounded.
There is $c>0$ such that there is no $c\text{-net}$ for $\overline{S}$.
But there must a $c\text{-net}$ for ${S}$.
So some $y\in\overline{S}$ such that $y$ is not in $\bigcup\limits_{k = 1}^n {B(\alpha _k ;c)}$ the $c\text{-net}$ for $S$.

Let $d=\min\{d(y,\alpha_k)-c\},~k=1,2,\cdots n$.
You need to argue that $d>0$.
But $B(y;d)$ must contain a point in $S$.
Now prove that is a contradiction.