Definition. If $p$ is a prime, then a p-group is a group in which every element has order a power of $p$. Remark: An additively writen group is called $bounded$ if its elements have boundedly finite orders. Of course multiplicative groups with this property are said to have finite exponent but this term is inappropriate in the context of additive groups.
Let $G$ be an infinite abelian p-group of bounded order, then prove that $G\cong \mathbb{Z}_{p^{n}}\oplus\mathbb{Z}_{p^{n}}\oplus H$, for some natural number $n$ and for some abelian group $H$.