 First we prove that none of the sentences \(S_i\) can be true. Assume to obtain a contradiction that \(S_i\) is true for some \(i\). Then it is true that “for all \(j\gt i, S_j\) is not true”. Thus none of the sentences \(S_j\) for \(j\gt i\) are true. In particular, \(S_{i+1}\) is not true. \(S_{i+1}\) is the sentence “for all \(j\gt i+1, S_j\) is not true”. Since this sentence is not true, there must be some \(k\gt i+1\) for which \(S_k\) is true. However, this contradicts that none of the sentences \(S_j\) with \(j\gt i\) are true.