Let $1\leq p<q<+\infty$. Let $B_n=\{⟨x_k\rangle\in l^q: \sum_k |x_k|^p\leq n\}$. Want to show: $B_n$ is closed and nowhere dense in $l^q$. Thus $l^p$ is of first category as a subset of $l^q$.
To show $B_n$ doesn't have interior points seems easy, since you can always construct a sequence of numbers that has arbitrarily small $l^q$-norm but diverges in $l^p$. But how to show it's closed? How to show that the $l^q$-limit of a sequence of elements in $l^p$ is also in $l^p$? I have no idea how to estimate the $l^p$-norm of the limit..
Partly for notational convenience and also partly to clarify that this isn't something special about $l^p$ and $l^q$, let us identify $l^p$ with $L^p(\mathbb{N},c)$ and $l^q$ with $L^q(\mathbb{N},c)$, where $c$ is the counting measure. For general $(X,\mu)$ we can prove the same statement for $L^p(X,\mu)$ and $L^q(X,\mu)$ as follows.
Let $B_n = \{ x \in L^q : \| x \|_p \leq n \}$. Suppose $x_m$ is a sequence in $B_n$ which converges in the sense of $L^q$ to $x$. To show $B_n$ is closed, it is enough to show that $x \in B_n$. To show $x \in B_n$, we need to show $x \in L^q$ and $\| x \|_p \leq n$. The first part follows because $x$ is the $L^q$ limit of a sequence.
To prove the second part, apply the $L^q$ convergence to extract a subsequence $x_{m_k}$ which converges almost everywhere. Now use Fatou's lemma to get that $\| x \|_p \leq \liminf \| x_{m_k} \|_p \leq n$.
So $x \in B_n$, and so $B_n$ is closed in the sense of $L^q$.