If $G=\langle\{a_i\}_{i=1}^r\rangle$ is nilpotent with $|a_i|=m_i$, show that $|G|$ divides a power of $\prod_{i=1}^rm_i$ and is finite.

Question

use this notation for the following $\textbf{Theorem}$ -

$\textbf{notation}-$ $G^n$ is the subgroup generated by $n$th power of elements of $G$

$\textbf{Theorem}$- In a finitely generated nilpotent group, $\cap{G^p}$, for any infinite set of primes, is finite,(follows by a paper of HIGMAN) does it help me to prove this question above. I don't think so.

Is it true, that a group generated by finite elements of finite order is finite. If it is true then to prove $G$ is finite in the problem is easy, and don not require nilpotency. But it should not be true, but what can be an example?