This is a specific question about the proof of Theorem 5.4 in "A Course in Functional Analysis" by Conway. The theorem is stated in the question.
The highlighted line confuses me. I thought it should be "ran U contains all functions f in l^2(E) such that f(e) = 0 for all but a COUNTABLE number of e". By Corollary 4.9 in the same book, if $E$ is an orthonormal set in $H$ and $h \in H$, then $<h, e> \neq 0$ for at most a countable number of vectors $e$ in $E$. If $H$ is infinite-dimensional (so its basis $E$ will also has infinite number of elements), and we pick $h$ as any element in its basis $E$, then we would have $f(e) := <h,e> = 0$ for all $e \neq h$ in $H$, i.e. infinitely many $e$'s.
The set A containing all functions f such that f(e) = 0 for all but a FINITE number of e is contained in the set B of "... but a COUNTABLE number ...", hence it is no contradiction to Corallary 4.9.
I think the author likes to use the "smaller" basis A, which is sufficient for his argumentation, but you could do the same argumentation with set B instead of A, as both A and B are dense in l^2(E).