I have a problem to proof this theorem, Anyone can help for detail.

"A subspace $\displaystyle Y$ of Banach space $\displaystyle X$ is complete if and only if $\displaystyle Y$ is closed in $\displaystyle X$"

I have an idea to prove this theorem, but I am not sure about this and I can't wrote it for detail.

Please correct my answer,

from left to right "let $\displaystyle X$ is Banach space, $\displaystyle Y\subset X$. so, $\displaystyle Y$ is Banach space. consider of Banach space definition, every Cauchy sequence of $\displaystyle Y$ is converge to $\displaystyle x\in X$ then $\displaystyle Y$ is closed on $\displaystyle X$".

right to left "I am still totally confuse..."