Separability of quotient in a Banach space

Question

I found a problem where it was requested to prove that if $X$ is separable Banach space and $Y$ a closed subspace of $X$, then the quotient $X/Y$ is also separable. What I cannot see clearly is wether the condition of $X$ being Banach is necessary or not. I sketched a proof of the problem and I did not use this condition at all. If anyone could guide me it would be really helpful. Thanks.

Answer 1

To reiterate what was stated in the comments, you can show more generally that if $X$ is a separable normed vector space and $Y$ is a closed subspace of $X$, then $X/Y$ is a separable normed vector space. So the hypothesis that $X$ is complete is not required to deduce that $X/Y$ is separable.