= Solution
The <Baire category theorem> says that a complete metric space cannot be a countable union of <nowhere dense set>[nowhere dense sets], equivalently that every countable intersection of open dense sets is dense. To prove it, let $G_1,G_2,\ldots$ be open dense subsets and let $V$ be nonempty and open. Choose nested closed balls
$$
\overline B(x_n,r_n)\subset B(x_{n-1},r_{n-1})\cap G_n,
\qquad 0<r_n<2^{-n},
$$
starting inside $V\cap G_1$. The centres are a <Cauchy sequence>; completeness gives a limit lying in every ball, hence in $V\cap\bigcap_nG_n$. Thus the intersection is dense.
Choose a countable sequence $q_j\uparrow p$ with $1\le q_j<p$. Since $\ell^q\subset\ell^r$ for $q<r$, every $\ell^q$ with $q<p$ lies in some $\ell^{q_j}$. For integers $m\ge1$, set
$$
E_{j,m}=\{x\in\ell^p:\|x\|_{q_j}\le m\}.
$$
Pointwise convergence and <Fatou lemma for series> show that $E_{j,m}$ is closed in $\ell^p$. It has empty interior: inside any $\ell^p$ ball, add a sufficiently long finite block whose $\ell^p$ norm is tiny but whose $\ell^{q_j}$ norm is larger than $m$. Therefore every $E_{j,m}$ is nowhere dense. Since $\ell^p$ is a <Banach space>, Baire gives
$$
\ell^p\ne\bigcup_{j,m}E_{j,m}
=\bigcup_{1\le q<p}\ell^q.
$$
An explicit element of the difference is
$$
\boxed{x_n=\frac1{n^{1/p}\{\log(n+1)\}^{2/p}}}.
$$
Its $p$th powers form a convergent logarithmic series, whereas for every $q<p$ the powers contain $n^{-q/p}$ with exponent below one and their series diverges. Hence $x\in\ell^p$ but $x\notin\ell^q$ for every $q<p$.
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