Does sequences in $\ell^p$ belong to some $\ell^q$ for $q

Does sequences in $\ell^p$ belong to some $\ell^q$ for $q

For every $p\in [1,\infty)$, consider the set $\ell^p$ of real sequences
$x=(x_n)=(x_1,x_2,\ldots)$ such that $\sum_{n=1}^\infty |x_n|^p<\infty$.
This is a Banach space with the norm $\Vert x\Vert_p=(\sum|x_n|^p)^{1/p}$.
I have the following problem:
Is it true that $\ell^p=\bigcup_{1\leq q<p}\ell^q$ for $p>1$?
I don't think it is true, but I'm having problems to show this. Clearly,
the statement is true for some $p>1$ iff it is true for every $p>1$.
I'm trying to find a sequence $(a_n)$ such that $\sum |a_n|^s$ converges
iff $s\geq 1$. I've tried many sequences (such as $(n^{1+1/n})$, involving
$\log$, etc...), but none of them worked.