Is it possible to show the existence of an infinite dimensional closed subspace of $\ell_p$ ($1<p<\infty$, $p\neq 2$), not isomorphic to $\ell_p$, in an **elementary** way?

For $1<p<q<2$, I think we can find such an example isomorphic to $\ell_p(\ell_q^n)$, but the proof I have in mind uses the fact that $\ell_q$ embeds in $L_p(0,1)$, which is not elementary.