  Choose some such asymptotically optimal function \(u\), and define the complexity (simpliciter) \(C(\sigma) = C_u (\sigma)\). Since \(u\) is optimal, it is near-superior to the identity function; it follows that there exists a \(k\) such that \(C(\sigma) \le \lvert\sigma\rvert + k\). On the other hand, we also know that the number of strings for which \(C(\sigma) \le \lvert\sigma\rvert - k\) is at most \(\frac{1}{2} ^k\). We know therefore that all except \(1 - 2^k\) sequences of length \(n\) have a complexity within \(k\) of \(n\). As \(n\) increases, for fixed large \(k\), therefore, we see that almost all sequences have complexity of approximately their length. All this can be used to make precise the definition of randomness sketched above.