I really like this one:
From Hölder, we get that the series $\sum_{i=1}^\infty x_iy_i$ always converges for $x \in \ell^p$ and $y \in \ell^q$. Since $\ell^p \subseteq \ell^q$ whenever $p < q$, we have that $\sum_{i=1}^\infty x_i^2$ also always converges. But this implies $\ell^p \subseteq \ell^2$ for any $p$.
Where is the problem?
Everything said is true, except for the last part; $\ell^p \subseteq \ell^2$ if, and only if $p \in (1,2)$. This is because, for the first series, $p$ and $q$ must be conjugates, and one conjugate is always in $(1,2)$ while the other is in $(2,\infty)$. To be able to say $\ell^p \subseteq \ell^q$, $p$ must be in the first interval.