These results may all be demonstrated by modifications of the diagonal argument by which Turing (1937) originally demonstrated the undecidability of the classical Halting Problem.[13] Nonetheless, Theorem 3.1 already has a number of interesting consequences about the relationships between the complexity classes introduced above. For instance, since the functions \(n^{k}\) and \(n^{k+1}\) satisfy the hypotheses of parts i), we can see that \(\textbf{TIME}(n^k)\) is always a proper subset of \