Let $A$ be an abelian $p$-group and let's denote $A[m]=\{a\in A\mid am=0\}$.
If there exists $n>1$ such that $$ A[p]\cap (pA[p^2])\cap (p^2A[p^3])\cap\dots\cap (p^{n-1}A[p^{n}])=\{0\} $$ does that implies that $A$ is finite?
This is the full problem that I am trying to solve, I already solved question a) and b) but I don't know how to solve part c). Would you please help me to do so. Thanks