3
$\begingroup$

I am trying to prove that the following is true:

Let $X$ be a Banach space, and $E$ a closed subspace, then $X/E$ is a Banach space

So the quotient norm is $\|[y]\|_{X/E} = \displaystyle\inf_{e\in E}\|y-e\|_X$. We want to show that an arbitrary Cauchy sequence in $X/E$ converges in $X/E$.

Let $\{Y_{n}\}_{n=1}^\infty$ be a Cauchy sequence in $X/E$.

Can we pick $y_n\in Y_n$ such that $\{y_n\}_{n=1}^\infty$ forms a Cauchy sequence, or is there some other approach? I have been having trouble thinking of an approach.

$\endgroup$