We have a metric space $\boldsymbol{(X,d)}$ and $\boldsymbol Y$ is included in $\boldsymbol{X}$, a subset.
How can we prove that $\boldsymbol Y$ is totally bounded if and only if the closure of Y is totally bounded?
Any help would be greatly appreciated.
The implication "$Y$ totally bounded then $\overline{Y}$ is totally bounded", is proved here among other places. (it's one of the "related" links on the right part of the web page; always good to pay attention to those..)
A subset of a totally bounded set is still totally bounded and this takes care of the other direction.