The only subgroup of finite order of a an infinite group is the identity

My proof

Let G be a group with infinite order

let H be a subgroup and let $\displaystyle x \in H $ and $\displaystyle x \ne 1 $

$\displaystyle \{x , x^2 , x^3 , x^4 , ... \} \subseteq H $

which make H with an infinite order

am I rite ?

is there any counter example