Suppose $p$ and $q$ are real numbers with $p^{-1}+q^{-1}=1$ and $p, q>1$. Show, quoting any results on convexity that you need, that

$$a^{1 / p} b^{1 / q} \leqslant \frac{a}{p}+\frac{b}{q}$$

for all real positive $a$ and $b$.

Define the space $l^{p}$ and show that it is a complete normed vector space.