Let $X$ be a metric space. Define what it means for a subset $E \subset X$ to be of first or second category. State and prove a version of the Baire category theorem. For $1 \leqslant p \leqslant \infty$, show that the set $\ell_{p}$ is of first category in the normed space $\ell_{r}$ when $r>p$ and $\ell_{r}$ is given its standard norm. What about $r=p$ ?