Riemannian covering completeness

Question

Let $\pi: (\tilde M, \tilde g) \to (M,g)$ a Riemannian covering. One knows that $M$ is complete if and only if $\tilde M$ is complete. This can be shown rather easily by lifting geodesics for example. My question is: how can one proof this theorem by showing metric completeness directly? One knows for example $d_{\tilde M} (x,y) \geq d_M(\pi(x),\pi(y))$, therefore $\tilde M$-Cauchy sequences are also $M$-Cauchy sequences. But how to proceed?