Exhaustive filtration of a torsion abelian group

Question

Let $p$ be a prime integer and $A$ a torsion abelian group. Define a filtration on $A$ by $F_{-1}A=0$ and $F_s A=\ker ( p^{s+1}:A\to A)$ for $s\geq 0$. Why is this filtration exhaustive, i.e. why is $A= \bigcup_s F_sA$.