Let $\displaystyle S_{\infty}$ be the group of all permutations of $\displaystyle \mathbb{N}$, viewed as a subspace of Baire space $\displaystyle \mathcal{N}$.

How do I prove that there is no left-invariant complete metric on $\displaystyle S_{\infty}$?

A metric is left-invariant if $\displaystyle d(xy, xz)=d(y, z)$, for all x, y, z.

Thank you.