Suppose $X$ is a totally bounded,complete metric space.Suppose $X$ is not compact.Then $\exists $ an open cover $(G_\alpha)$ of $X$ with no finite subcover.Now $X$ is totally bounded.So,it has a finite cover by $1$-balls in $X$.Now,at least one of these balls is not covered by finitely many $G_\alpha$'s because otherwise $(G_\alpha)$ will have a finite subcover.Suppose the ball is $B_1$.Now $B_1$ being a subset of $X$,is totally bounded.So,it has a finite cover by $1/2$-radius open balls in $X$,say $B(x_1,1/2),...,B(x_k,1/2)$Now,at least one of $B(x_i,1/2)\cap B_1$ cannot be covered by finitely many $G_\alpha$'s.Call the corrsponding ball $B_2$,then it is also totally bounded,then it has a finite cover by $1/3$-balls of $X$.Again we give a similar argument and continuing this process,we can get hold of a sequence $(B_n)$ .Notice that $B_1\supset B_1\cap B_2 \supset B_1\cap B_2\cap B_3\supset\dots$.Define $A_n=B_1\cap B_2\cap\dots\cap B_n$,notice that $A_n$ cannot be covered by finitely many $G_\alpha$'s.Then $A_1\supset A_2 \supset A_3\supset\dots$,then $\overline A_1\supset \overline A_2\supset\dots$.Where $0\leq diam(\overline A_n)= diam(A_n)\leq diam(B_n)\leq 1/n \to 0$,so,$diam(\overline A_n)\to 0$.So,Since $X$ is complete,so by Cantor's intersection theorem $\cap_{1}^{\infty}\overline A_n$ is non-empty and is a singleton $\{x\}$ say.Now,$\{x\}$ can be covered by only one $G_\alpha$.Now $x\in G_\alpha$ and this set is open,so $\exists m \in \mathbb N$ such that $B(x,1/m)\subset G_\alpha$.Take $A_{2m}$.Let $y\in A_{2m}$,then $y\in \overline A_{2m}$.Also $x\in \overline A_{2m}$,then $d(x,y)\leq 1/2m<1/m\implies y\in B(x,1/m)$,So,$A_{2m}\subset B(x,1/m)\subset G_\alpha$,but $A_{2m}$ cannot be covered by finitely many $G_\alpha$'s,thus we have a contradiction.
I have done this proof on my own,but I am not completely sure that it is alright,I know that the idea is correct.But it may have some little subtle mistakes or errors that should be removed by modification.Can someone please tell me whether I have missed some logic or done something wrong in any step or it is completely correct?