Do sequences in $\ell^p$ belong to some $\ell^q$ for $q<p$?

Question

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.

Answer 1

No. Consider $$a_n=\frac{1}{(n(\ln n)^2)^{1/p}}.$$