This paper deals with the proof of the existence of kink states in the discrete model of the polyacetylene molecule . We use ideas from Kennedy and Lieb to study finite, odd chains of polyacetylene, and then we consider the limit as the number of atoms goes to infinity. We show that, after extraction of a subsequence and up to a translation, the energy minimizers of odd chains tend to an infinite vector approaching one of the infinite dimerized states at plus infinity and the other one at minus infinity. This state is called a kink and its existence was strongly suggested in several works in the physical literature, but a mathematical proof was missing, to our knowledge.