LetAbe a PI-algebra with generators {ai},i= 1,…k, andWnbe a set of all words in {ai} with length less or equaln, wherenis PI-degree ofA. If all elements ofWnare algebraic, thenAis finite dimensional. This theorem was firstly proved by V. Ufnarovsky, but the proof was too difficult. In this paper we give the short one. It based on the upper boundary for the height of algebraA, and on the consideration of infinite words (superwords) in algebras.