Free subgroups of the group of infinite unitriangular matrices

In this note we give simple explicit examples of free subgroups of rank 2 in the group of infinite upper unitriangular matrices over integers. The proofs that the given subgroup is free are elementary. Using canonical projections mod p (p - any odd prime) we obtain also free subgroups in the group of finite state automata on alphabet of size p, extending results of Aleshin [1], Brunner-Sidki [2] and Olijnyk-Sushchansky [8, 9] in this direction. As application, we give the simple proofs of two classical results on free groups.