Inclusion of $l^p$ space for sequences
If $\sum_n |x_n|^p < \infty$, then $|x_n| \leq 1$ definitely. Therefore, if $q>p$, then $|x_n|^q \leq |x_n|^p$, and we conclude that $\ell^p \subset \ell^q$.
Now the opposite embedding can't be true, otherwise $\ell^p \simeq \ell^q$ for every pair $(p,q)$, and this is obviously false.
For completeness, a counterexample for $p>q$: pick $s\in (1/p,1/q)$ and consider the sequence $x_n=1/n^s$. This sequence is in $\ell^p$ but not in $\ell^q$.