= Solution
Let $q=p/(p-1)$. The scalar <Young inequality> $ab\leq a^p/p+b^q/q$ follows by maximizing $ab-a^p/p$ over $a\geq0$, obtaining $b^q/q$. If both norms are nonzero, apply it to $|x_j|/\|x\|_p$ and $|y_j|/\|y\|_q$, and sum over a finite initial segment. Passing to the increasing limit gives
$$
\boxed{\sum_{j=1}^\infty|x_jy_j|\leq\|x\|_p\|y\|_q.}
$$
Zero norms give the trivial case. This proves <Hölder's inequality> without assuming convergence of the product series in advance.
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