Separability of Normed Spaces with Schauder Bases

Let $(x_n){n/in/mathbb{N}}$ be a Schauder basis for the normed space $X$. For each $n/in/mathbb{N}$, let $q_n$ be a rational number such that $/|x_n-q_nx_n/|</frac{1}{2^n}$. Then the set $D=/{q_nx_n:n/in/mathbb{N}/}$ is countable because it is indexed by $/mathbb{N}$, and we claim that $D$ is dense in $X$./n/nTo see this, let $x/in X$ and let $/epsilon>0$. Since $(x_n)$ is a Schauder basis, there exist scalars $(a_n)$ such that $/|x-/sum{n=1}^N a_nx_n/|</frac{/epsilon}{2}$ for some $N/in/mathbb{N}$. Then/n/begin{align*}/n/|x-q_Nx/|&/leq/|x-/sum_{n=1}^N a_nx_n/|+/|/sum_{n=1}^N a_nx_n-q_N/sum_{n=1}^N a_nx_n/|+/|q_N/sum_{n=1}^N a_nx_n-q_Nx/| // /n&/leq/frac{/epsilon}{2}+/|/sum_{n=N+1}^/infty a_nx_n/|+/|q_N/sum_{n=N+1}^/infty a_nx_n/| // /n&/leq/frac{/epsilon}{2}+/sum_{n=N+1}^/infty |a_n|/|x_n/|+/frac{1}{2^N}/|/sum_{n=N+1}^/infty a_nx_n/| // /n&/leq/frac{/epsilon}{2}+/frac{/epsilon}{2}+/frac{/epsilon}{2} // /n&=/epsilon/n/end{align*}/nTherefore, $D$ is dense in $X$, and hence $X$ is separable.