An operator is compact only if it maps weakly convergent sequences into strongly convergent sequences.

Question

I have a problem with the proof of the following statement:

A linear operator $A$, defined on a Hilbert space $H$, maps any weakly convergent sequence $(x_n)$ into a strongly convergence sequence $(Ax_n)$ only if it is compact.

I found a proof in "Introduction to Hilbert Spaces with Applications" by Lokenath Debnath and it starts with words "Let $(e_n)$ be a complete orthonormal sequence in $H$,..."

I do not understand why there is such a sequence because $H$ is not necessarily separable. Could you please explain why such a sequence exists or suggest some other book.