Every infinite G has a S.T $o(a)=\infty?$ [closed]

Is it true to claim that for every infinite Group G then there is a in G such that $o(a)=\infty?$

I took few examples and this sound correct, but any ideas on how can I prove this in general?

It's not true.

For a counterexample, the direct sum of infinitely many cyclic groups of order $2$ is an infinite group, but every element of that group has order 2.

One could restrict the question the way that Burnside restricted it in the early 20th century: if a group $G$ is infinite and finitely generated, does it have an element of infinite order? Even that turns out to be false but it was much harder to find a counterexample. You can read about counterexamples in that link.