Hölder's inequality

Young's inequality. For $x,y\ge0$ and $\frac1p+\frac1q=1$:
$$xy\le\frac{x^p}{p}+\frac{y^q}{q}.$$
This follows from concavity of $\ln$ (weighted AM-GM).

Normalization. Assume wlog that $\sum a_i^p=\sum b_i^q=1$ (the general case follows by scaling). Then we need to show $\sum a_ib_i\le1$.

Apply Young. $a_ib_i\le\frac{a_i^p}{p}+\frac{b_i^q}{q}$. Summing:
$$\sum a_ib_i\le\frac1p\sum a_i^p+\frac1q\sum b_i^q=\frac1p+\frac1q=1.$$

Conclusion. Hölder's inequality holds. For $p=q=2$, it reduces to Cauchy-Schwarz.