The problem is to prove that every Hilbert space has a orthonormal basis. We are given Zorn's Lemma, which is taken as an axiom of set theory:
Given a orthonormal set $E$ in a Hilbert space $H$, it is apparently possible to show that $H$ has an orthonormal basis containing $E$.
I tried to reason as follows: Suppose $E$ is a finite set of $n$ elements. Then one can number the elements of $E$ to create a totally ordered set of orthonormal elements. Then the span $<E>$ can be identified with $R^n$, where each element $v = v_1 e_1 + v_2 e_2 + ... + v_n e_n$ is identified with the vector $(v_1, v_2, ..., v_n)$. On $R^n$ we have a total order, namely the "lexigraphical order" $(x_1,x_2,...,x_n) \leq (y_1,y_2,...,y_n)$ if $x_1 < y_1$ or if $x_1 = y_1$ and $x_2 < y_2$ or if $x_1 = y_1$, $x_2 = y_2$ and $x_3 < y_3$ and so on. Hence $E$ is a totally ordered subset of $H$ and $H$ is a partially ordered set. However, this set doesn't seem to have an upper bound. The set $E$ does have an upper bound. If we define a total order on $E$ only, then X is a partially ordered set satisfying the criteria so X has a maximal element?
This is as far as I got, and I am not sure the entire argument is correct. I don't see how what kind of maximal element I am seeking, since the orthonormal basis of a Hilbert space can have countably infinity number of elements.