Let $X_i$ be family of normed linear spaces(i=1,2.... )Let $X=\{(x_i)\mid x_i\in X_i\}$ and $\sum\|x_i\|^{p}_{X_i} < \infty$. Put a norm on $X$ by $\left(\sum\|x_i\|^{p}_{X_i}\right)^{1/p}$ where p$\geq$1 . Then $X$ is a Banach space if and only if each $X_{i}$ is a Banach space.
I got the converse whose argument is quite similar to how we showed $l_{p}$ is Banach but I am stuck on implies part.
I'd like to have some help about that.