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

"A subspace Y of Banach space X is complete if and only if Y is closed in 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 X is Banach space, Y\subset X. so, Y is Banach space. consider of Banach space definition, every Cauchy sequence of Y is converge to x\in X then Y is closed on X".
right to left "I am still totally confuse..."