$C_{c}(X)$ is complete implies $X$ is compact.

Question

If $X$ locally compact Hausdorff space. Then $C_{c}(X)$ is complete implies $X$ is compact.

I know that $C_{c}(X)$ dense in $C_{0}(X)$. So in that case $C_{c}(X)=C_{0}(X)$. I know only Tiez extension theorem on locally compact Hausdorff space and vary basic knowledge of Banach Algebra.