I have the following problem$^*$:
Prove that the group $G$ having generators and relations respectively $X=\{x_0,x_1,x_2,\ldots\} \\\{px_0=0,x_0=p^nx_n, \text{all } n\geq1\}$ is an infinite $p$-primary group with $\bigcap_{n=1}^\infty p^nG\neq0$.
What I have done:
Since the set of generators are infinite so is the group. For another claim, I suppose $F$ to be free abelian group on $X$ and let the subgroup of it, say $R$ generated by the relations. Now $a_n=x_n+R\in G$ and $pa_0=0, p^2a_1=pa_0=0,p^3a_2=p^2x_0=0,\ldots$ and so $p^{n+1}a_n=0$. I conclude the group is $p$-primary. For the last I see that $x_0$ is an element in that intersection.
Is my approach right? Did I conclude correctly that the above intersection has $x_0$ in itself or the intersection can have other elements than $x_0$? Thanks.
$(*)$ : An Introduction to the Group Theory by J.J.Rotman pp. 317 problem 10.5