Definition of Separable Space The standard definition (e.g. from wikipedia) that a separable topological space $X$ contains a countable, dense subset, or equivalently that there is a sequence $(x_n)$ of points in $X$ such that every point $y \in X$ is $\epsilon$ close to some point in $(x_n)$. I can see that these two definitions are equivalent since a countable set has a bijection with the natural numbers, and so the countable base can be expressed as a sequence $(x_n)$.
However, in Zettili's Quantum Mechanics: Concepts and Applications, the definition of the separability property for a Hilbert space $H$ reads:

There exists a Cauchy sequence $\psi_n \in H$ ($n = 1, 2, ...)$ such that for every $\psi$ of $H$ and $\epsilon > 0$, there exists at least one $\psi_n$ of the sequence for which $$||\psi-\psi_n|| < \epsilon.$$

How is it without loss of generality to restrict the "countable basis" sequence to be Cauchy? I am having a hard time coming up with such a Cauchy sequence even for $\mathbb{R}$.
 A: Zettili misspoke, I think. Yes, there will be a sequence $(\phi_n)_n$ from $H$ such that for all $\phi \in H$ and all $\varepsilon >0$ we can find some $n$ such that $\|\phi_n - \phi\| < \varepsilon$, which is just a restatement of the general topology concept of separability in the Hilbert space context.
What we cannot have that this sequence will be Cauchy. If it would be it would converge to some $\phi_0 \in H$ and that kills the approximation property. He must have meant something else.
For Hilbert spaces we can also say that separability in an infinite dimensional Hilbert space implies there is an orthonormal basis $(e_n)_n$ which means

*

*$\langle e_n, e_m \rangle = 0$ for all $n \neq m$.

*$\|e_n\|=1$ for all $n$.

*For all $x \in H$ there is a sequence $(c_n)$ of scalars (usually complex) such that $x = \sum_n c_ne_n$, convergence is in the norm.

The latter form you will probably also meet. It follows from a countable dense set  plus the extra structure a Hilbert space has compared to a "mere" topological space.
