Nilpotency of fixed class is quotient-closed

Statement
Suppose $$G$$ is a nilpotent group and $$H$$ is a normal subgroup of $$G$$ and $$G/H$$ is the corresponding quotient group. Then, $$G/H$$ is also a nilpotent group and its nilpotency class is at most equal to the nilpotency class of $$G$$.

Note that we say that a group is nilpotent "of class $$c$$" if its nilpotency class is at most $$c$$. The statement can thus be reformulated as saying that the property of being nilpotent of class $$c$$ is closed under taking quotients.

Related facts

 * Nilpotency of fixed class is subgroup-closed
 * Nilpotency of fixed class is varietal
 * Nilpotency of fixed class is direct product-closed