# Thread: left-invariant complete metric on S_\infty

1. ## left-invariant complete metric on S_\infty

Let $S_{\infty}$ be the group of all permutations of $\mathbb{N}$, viewed as a subspace of Baire space $\mathcal{N}$.
How do I prove that there is no left-invariant complete metric on $S_{\infty}$?

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

Thank you.