Equivalence of definitions of nilpotent variety

The two definitions that we have to prove as equivalent
Consider a subvariety of the variety of groups, i.e., a collection of groups (closed upto isomorphism) that is closed under taking subgroups, quotients, and arbitrary direct products. Then, the following are equivalent:


 * 1) Every group in the variety is a nilpotent group
 * 2) There exists a constant $$c$$ such that every group in the variety is nilpotent with nilpotence class at most $$c$$.

Proof
Clearly, (2) implies (1). We need to show that (1) implies (2). We do this by contradiction: suppose every group in the variety is nilpotent, and suppose that, for every $$c$$, there exists a $$G_c$$ in the variety with nilpotence class strictly greater than $$c$$. Then, let $$G$$ be the direct product of all the $$G_c$$s.

Suppose the group $$G$$ is nilpotent of nilpotence class $$d$$. Then, every subgroup of $$G$$ has nilpotence class at most $$d$$. But, $$G$$ has a direct factor $$G_d$$ of nilpotence class more than $$d$$, leading to a contradiction.