I am trying to solve problem 5.45 from Rotman's book:
For each $n\geq 1$ , let $G_n$ be a finite $p$-group of class $n$. Define $H$ to be the group of all sequences $(g_1, g_2, ... )$, with $g_n\in G_n$ for all $n$ and with $g_n = 1$ for all large $n$; that is, $g_n \neq 1$ for only a finite number of $g_n$. Show that $H$ is an infinite $p$-group which is not nilpotent.
My attempt: $H= Dr_{n\in \mathbb{N}} G_n$ , or in the Rotman's book we show $H= \sum_{n\in \mathbb{N}} G_n$ which we call direct sum. I am thinking about what can be said about the nilpotency (or lack of nilpotency) of subgroups of a nilpotent group of class $c$, and assuming, by a contradiction that $H$ is nilpotent.
I am just not sure how to proceed here. Any help would be appreciated.