My question is simple.

Subgroup of a torsion-free group is torsion-free again.

True or false?

A group is called torsion-free if every element in has infinite order except the identity.

We know that infinite group has infinite number of subgroups.

But are the subgroups of infinite order too?

So, for my question here, is it true that all the elements in any subgroup of torsion-free group has infinite order?