Question about finite group

Question

We consider $A$ an abelian $p$-group and we 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?

Answer 1

If the group $A$ has finite exponent, that is, $p^mx=0$ for every $x$, then $$ p^mA[p^{m+1}]=\{0\} $$ Consider an infinite direct sum of copies of $\mathbb{Z}/p\mathbb{Z}$ and you have a counterexample.