Solution
= Solution
Apply <Hölder's inequality> to the finite sum with the second sequence $|x_j+y_j|^{p-1}$:
$$
\sum_{j=1}^N|x_j+y_j|^p
\leq(\|x\|_p+\|y\|_p)\left(\sum_{j=1}^N|x_j+y_j|^p\right)^{(p-1)/p}.
$$
Divide by the last power if the sum is nonzero. The resulting bound is independent of $N$, so passing to the limit proves both membership in $\ell^p$ and the <Minkowski inequality>
$$
\boxed{\|x+y\|_p\leq\|x\|_p+\|y\|_p.}
$$