A Banach space with a Schauder basis is separable

In the real case let $D$ consist of all finite linear combinations of the basis vectors with rational coefficients. In the complex case use coefficients in $\mathbb{Q}+i\mathbb{Q}$. This is a countable union of countable finite products and hence is countable. (definition).

Given $x$, first use [L1] to choose a basis partial sum within $\epsilon \mathrm{/}2$ of $x$. Approximate its finitely many scalar coefficients by rational, respectively Gaussian-rational, scalars closely enough that the resulting finite combination changes by less than $\epsilon \mathrm{/}2$. It belongs to $D$ and is within $\epsilon$ of $x$. Thus $D$ is dense. [L1, step 1.1, finite approximation] (finite approximation).