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.