Necessary and sufficient condition for $l_p$ space inclusion

Question

I am trying to prove the following statement.

Given $1<p,q<\infty$, $p<q$ iff $l^p \subset l^q$.

The forward direction is easy. I having trouble with the opposite direction and I don't know where to start. Can someone help me with this?

Answer 1

Assuming that $p>q$ and $\ell^{p}\subset\ell^{q}$, choose some $r>1$ with $rq<p$, then the sequence $(n^{-r/p})_{n}$ is in $\ell^{p}$ but $\sum\dfrac{1}{n^{(rq)/p}}=\infty$, so $(n^{-r/p})_{n}\notin\ell^{q}$, a contradiction.