Nilpotency of fixed class is direct product-closed

Version in terms of fixed class bound
Suppose $$G_i, i \in I$$ is a collection of groups indexed by an indexing set $$I$$. Suppose there is a positive integer $$c$$ such that each $$G_i$$ is a nilpotent group of nilpotency class at most $$c$$.

Then, the external direct product of the $$G_i$$s is also a nilpotent group of nilpotency class at most $$c$$.

Version in terms of maximum class
Suppose $$G_i, i \in I$$ is a collection of groups indexed by an indexing set $$I$$. If all the $$G_i$$s are nilpotent groups and there is a common finite bound on their nilpotency class values, then the external direct product of the $$G_i$$s is also a nilpotent group and its nilpotency class is the maximum of the nilpotency class values of all the $$G_i$$s.

In particular, for two nilpotent groups $$G_1$$ and $$G_2$$ of nilpotency classes $$c_1,c_2$$ respectively, the nilpotency class of $$G_1 \times G_2$$ equals $$\max \{ c_1, c_2 \}$$.

Related facts

 * Solvability of fixed length is direct product-closed
 * Central series is direct product-closed
 * Upper central series is direct product-closed
 * Lower central series is direct product-closed