Is $c$, the Banach space of convergent sequences with the sup norm, separable?
Let $X$ be the set of all sequences which are rational numbers, that converge to some rational number $x$. As the rationals are countably infinite, we need to only show that $X$ is dense in $c$.
That is what I am a bit unsure how to show… help would be appreciated.
Best Answer
Hint. Consider the set $S$ of sequences of rational numbers that are eventually constant.