I've proved that every totally bounded set is bounded and if a subset of $\mathbb{R^n}$ is bounded then it is totally bounded (wrt every metrics and norms).
Also, we have given this example below at lesson.
$\mathbb N$ is bounded wrt $d=\min\{|x-y|,1\}$ but it is not totally bounded because there is no finite $\varepsilon$- net for $\varepsilon=1/2$. However, $\mathbb N \subset \mathbb R$ It must have been totally bounded?
It can be so nonsencial. Forgive me for this. I cannot see my mistake.
I appreciate any help. Thanks in advance
It is no true that bounded sets are totally bounded w.r.t. any metric on $\mathbb R^{n}$. This is true for the usual metric but not for general metric.
For the discrete metric no infinite set is totally bounded but the whole space is bounded.
However, all norms on $\mathbb R^{n}$ are equivalent and totally bounded sets in a norm are nothing but bounded sets since compact sets are the same for any two norms.