Equivalence of definitions of solvable variety

The 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 direct products. Then, the following are equivalent:


 * 1) Every group in the variety is a solvable group
 * 2) There exists a nonnegative integer $$l$$ such that every group in the variety is solvable with solvable length at most $$l$$

Proof
Clearly (2) implies (1). We need to show that (1) implies (2), which we do by contradiction. Suppose there exists a variety of groups where every group is solvable, but where, for every $$l$$, there exists a member $$G_l$$ of the variety that has solvable length strictly greater than $$l$$.

Now, consider $$G$$ to be the direct product of the $$G_l$$s. Then, $$G$$ is in the variety (since a variety is closed under arbitrary direct products). Suppose $$G$$ is solvable. Let $$m$$ be the solvable length of $$G$$. Then, every subgroup of $$G$$ has solvable length at most $$m$$. But $$G$$ has as a direct factor (and hence as a subgroup) the subgroup $$G_m$$, whose solvable length is strictly greater than $$m$$, a contradiction.