Chapter 4.7, Problem 11E

Assume that $H_1,H_2,\mathrm{...},H_n$ are subgroups of the abelian group $G$ such that the sum $H_1+H_2+\mathrm{...}+H_n$ is direct. If $K_i$ is a subgroup of $H_i$ for $i=1,2,\mathrm{...},n,$ prove that $K_1+K_2+\mathrm{...}+K_n$ is a direct sum.