If the orders of all elements of a group are uniformly bounded by some finite number, is every finitely generated subgroup finite?

Question

Clearly this is equivalent to there existing an $n\in \mathbb{N}$ such that $a^n=e$ for all elements $a$ in our group, say $G$. Given a symmetric finite set $\{a_1, a_2, \ldots ,a_m\}$ containing the identity, the subgroup it generates then coincides with all words in the $a_i$'s. Using that $a^n=e$ for all $a\in G$ we can then try to reduce the length of such words, but is it possible to do so in a way that uniformly bounds the length of the new words?