I think for some time and now I agree that the statement is false. But I cannot prove or disprove the statement.
I know that compact $\Leftrightarrow$ complete+totally bounded, and also "bounded and totally bounded are the same in $\Bbb R^n$".
So I aim to find a not complete metric space which is closed and totally bounded. Could someone please help? If such an example exists, may I please ask for finding one in $\Bbb R^n$ (or other easy examples are also appreciate).
If the statement is correct, may I please ask for a proof?