Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 22H / a / Solution

Let $q=p/(p-1)$. The scalar Young inequality $ab\le a^p/p+b^q/q$ follows by maximizing $ab-a^p/p$ over $a\ge 0$, obtaining $b^q/q$. If both norms are nonzero, apply it to $|x_j|/\Vert x{\Vert }_p$ and $|y_j|/\Vert y{\Vert }_q$, and sum over a finite initial segment. Passing to the increasing limit givesZero norms give the trivial case. This proves Hölder's inequality without assuming convergence of the product series in advance.