Consider an execution of the Gale$-$Shapley algorithm in which $n$ candidates are applying for $n$ jobs. All the candidates have ranked all the companies from $1$ to $n$ and all the companies have ranked all the candidates from $1$ to $n.$ The candidates are proposing, and the companies are accepting or rejecting the proposals.

Consider the following statement: There is at most one applicant who gets their last choice. Either prove that this statement is correct or give a counterexample.