TITLE: Cartesian Product of Boundedly Finite Groups
QUESTION [5 upvotes]: Let $G$ be the cartesian product of countably many finite groups $H_\alpha$,  $\alpha\in \omega$. Assume also that there exists an $ n\in\mathbb N$ such that $|H_\alpha|\leq n,\, \forall \alpha\in \omega$. Is $G$ locally finite? Is there an easy way to prove it?
Notation
A group $G$ is said to be locally finite iff every finitely generated subgroup is finite.

REPLY [2 votes]: There are only finitely many groups of order at most $n$, let $H$ be the direct product of all of these, then $H$ is still finite and we have embeddings $H_\alpha\hookrightarrow H$ for all $\alpha\in\omega$. Let $X = \prod_{\alpha\in\omega} H$, then $G\hookrightarrow X$, so it suffices to show that the $X$ is locally finite.
Thus we have reduced the claim to the special case of all $H_\alpha$ being the same finite group.
Let me know if this helps.