3 lemmas of class 28

Lemma 23.3

For set $S\subset X$, $S$ is totally bounded $⟹$ $S$ is bounded.
Suppose $S$ is totally bounded, we have

$$(\forall \epsilon >0)(\exists m)(\exists x_1,x_2,\dots ,x_m\in X)\bigcup _{\alpha =1}^{m}S\subset B_{\epsilon }(x_{\alpha })$$

and we want to prove that

$$(\exists R>0)(\exists x\in X)S\subset B_R(x)$$

Then take $\epsilon =1$ and take $x=x_{\alpha }$ where $\alpha \le m$, we can take $R=\max (\{d(x_{\alpha },x_k)\mid x_k\in X\})$ which is the max distance between every points in $X$.
After that, it’s safe to say $S$ is bounded. $\square$

Lemma 23.4

For set $S\subset {\mathbb{R}}^d$, $S$ is bounded $⟹$ $S$ is totally bounded