It should be clear that it suffices to prove the following:
Let an orthonormal set in a Hilbert space , then is a Hilbert base ( ) we have
Since is a Hilbert base, we have and so, we divide this in two cases:
1) : We have then where (where is the basefield or ) then . Thus , and as such for every we have and so
2) : We choose a such that . Then . Now we take and using the triangle inequality two times and the Cauchy-Schwarz inequality afterwards we obtain:
Thus we have shown that
Since , we are finished.
Since for all we have then , we have that and so, , and so .
Man, that was hard to type, anyway hope it helps.