Confused about part of proof of showing what the completion of the direct sum of Banach spaces is

Question

For notation $P_j$ is the projection maps a more common notation is $\pi_j$

I am a bit confused about why $\lim ||x_n||_p^p$ converges? Would I use the fact that $P_j(x_n)$ is Cauchy and it would be some computational epsilon proof? How does showing that every finite subset $I$ of $J$, $\sum_I ||\tilde{x}_j||^p < \infty$ show that $\sum_J ||\tilde{x}_j||^p < \infty$?