I am trying to prove the theorem: A normed space $X$ with a Schauder basis is separable.
I tried to do it the following way.
Let $X$ be a real space. Suppose that $(e_n), e_n\in X\,\forall n\in \mathbb N$ be a Schauder basis for $X$.
Let $A:=\{\sum_{i=1}^n r_i e_i: r_i\in \mathbb Q, n\in \mathbb N\}$. $A$ is countable: for a given $n$, the linear combination $\sum_{i=1}^n r_i e_i$ can be written in $\aleph^n=\aleph$ ways as $r_i\in \mathbb Q$ and the cardinality of $\mathbb Q=\aleph$. So $A$ is a countable union of such linear combinations (countably many) hence $A$ is countable.
I claim that $A$ is dense in $X$. To prove that it is so, take any $\epsilon>0$ and any $x\in X$. It is enough to show that there is some $a\in A$ such that $\|x-a\|<\epsilon.$
By definition of Schauder basis, there exists $N\in \mathbb N, (c_i)_{i=1}^\infty\subset \mathbb R$ such that $\|\sum_{i=N+1}^\infty c_ie_i\|\lt \frac \epsilon 2.$
Using density of rationals, choose $r_i\in \mathbb Q$ such that $\sum_{i=1}^N\| (-r_i+c_i)e_i\|\lt \frac \epsilon 2$.
Setting $a:= \sum_{i=1}^N r_i e_i$, it follows that $$\|x-a\|=\|\sum_{i=1}^N(-r_i+c_i)e_i+\sum_{i=N+1}^\infty c_ie_i\|\le \|\sum_{i=1}^N(-r_i+c_i)e_i\|+\| \sum_{i=N+1}^\infty c_ie_i\| <\epsilon.$$
QED.
Please let me know in case there is any error in it. Thanks a lot for your time.