An abelian group $G$ has property $N$ if for every sequence of subgroups of $G$ such that $G_1 \subseteq G_2 \subseteq ... \subseteq G$, then there exists positive integer $N$ such that $G_N=G_M$ for all $M \geq N$. Now let $\phi:G \rightarrow G_1$ be a group homomorphism of abelian groups. Prove that if $G$ has property $N$, then $\ker(\phi)$ has property $N$.
I know that given any sequence $K_1 \subseteq K_2 \subseteq ... \subseteq K \subseteq G$ there is an integer $N$ satisfying the above property since $G$ has property $N$. But how do I show that when we leave out $G$ this is still true? Any hints? Thanks.