Following an old question (link at the bottom) that was never answered, prove that if $\displaystyle (X,d)$ is a non-compact metric space then there is a metric $\displaystyle d^*$ equivalent to d such that $\displaystyle (X,d^*)$ is not complete. In other words all equivalent metrics on $\displaystyle X$ are complete if and only if $\displaystyle X$ is compact.

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