is the union of two totally bounded sets totally bounded?

Question

i am trying to prove that the union of A and B is totally bounded when A and B are both totally bounded. Firstly , A is totally bounded so there is a e-dense subset so that A subset of`$U_{i=1}^{n}B\left( x_{i},δ \right)$ and B is a subset of $U_{j=1}^{m}B\left( x_{j},\varepsilon \right)$ so A union B is a subset of what? I am kind of stuck . Should i set d=max of e and δ ?

Answer 1

Take $\varepsilon>0$. Since $A$ is totally bounded, there are points $a_1,\ldots,a_n\in A$ such that $A\subset\bigcup_{j=1}^nB(a_j,\varepsilon)$. And, since $B$ is totally bounded, there are points $b_1,\ldots,b_m\in B$ such that $B\subset\bigcup_{j=1}^mB(b_j,\varepsilon)$. But then$$A\cup B\subset\bigcup_{p\in\{a_1,\ldots,a_n,b_1,\ldots,b_m\}}B(p,\varepsilon).$$