I was doing some problems from Rudin's Principles of Mathematical Analysis and came across a problem in which he asks you to prove Hölder's inequality via Young's inequality:
If $u$ and $v$ are nonnegative real numbers, and $p$ and $q$ are positive real numbers such that $\displaystyle \frac{1}{p}+\frac{1}{q}=1$, then $\displaystyle uv \leq \frac{1}{p}u^p+\frac{1}{q}v^q$.
I'm familiar with the proof using convexity of the $\log$ function and Jensen's inequality, but Rudin hasn't defined the $\log$ function by chapter $6$ (where this problem originates) and hasn't done anything with convexity. Usually he gives everything necessary for a problem before he poses one, so this seems to be something of an omission. Perhaps he wants us to read Chapter $8$ to learn about $\log$ and prove Jensen's inequality before attacking this problem? But then why put it in Chapter $6$?
My question: is there a proof of Young's inequality that does not use convexity of $\log$ or something similar? If one exists, can it be done using only the material from chapters $1-6$ of Principles?
(For clarity, chapter 1-6 essentially cover the real number system, metric space topology, sequences and series, continuity, differentiability, and the Riemann-Stieltjes integral.)