Background:
Let $E,F$ be Banach spaces with $F$ be finitely representable in $E$, and separable. That is, for all finite dimensional subspaces $M\subset F$ there is a $(1 + \epsilon)$-isomorphism (not necessarily surjective) $T:F\to E$.
Note: A $(1+\epsilon)$-isomorphism $T:M\to F$ is a linear operator such that $\|T\|,\|T^{-1}\|< (1+\epsilon)$, where $T^{-1}$ is restricted to the image of $T$.
Question:
How can I construct a linearly independent sequence $x_{n}\in F$ such that $\overline{\text{span}}\{x_{n}\} = F$?
Context:
I need to do so to follow the proof that under these hypotheses, $F$ isometrically embeds into $(E)_{\mathcal{U}}$ for all countably incomplete $\mathcal{U}$.
So, finite representability isn't even needed, but it is implicitly assumed that $F$ is infinite-dimensional.
Start with a countable dense subset $G\subset F$, which exists by separability.
Then choose a linearly independant sequence from elements of $G$, which must exist otherwise $F$ is finite-dimensional.