Nontrivial subgroup of nilpotent group has nontrivial homomorphism to center

Statement
Suppose $$G$$ is a nontrivial nilpotent group. Denote by $$Z(G)$$ the center of $$G$$. Suppose $$H$$ is a nontrivial subgroup of $$G$$. Then, there exists a nontrivial homomorphism of groups $$\varphi:H \to Z(G)$$.

Related facts

 * Nilpotent implies center is normality-large
 * Equivalence of definitions of characteristic direct factor of nilpotent group

Proof
The idea is to use an iterated commutator operation where all the other coordinates are fixed.