$S$ is a basis for it's own span if and only if it's a countable orthonormal set

Question

A section on Hilbert Spaces makes the following claim (item A):

Obvious if $S$ is orthonormal it must be linearly independent and thus be a base for the span. The converse doesn't seem true. For example i could pick two non-orthogonal vector that generate a plane.

I must be missing something here.