Summary: In 1965, Fine and Wilf proved the following theorem. if \((f_n)_{n\geq 0}\) and \((g_n)_{n\geq 0}\) are periodic sequences of real numbers, of period lengths \(h\) and \(k\), respectively, and \(f_n=g_n\) for \(0\leq n<h+k-\text{gcd} (h,k)\), then \(f_n=g_n\) for all \(n\geq 0\). Furthermore, the constant \(h+k-\text{gcd}(h,k)\) is best possible. In this paper, we consider some variations on this theorem. In particular, we study the case where \(f_n\leq g_n\) instead of \(f_n= g_n\). We also obtain generalizations to more than two periods. We apply our methods to a previously unsolved conjecture on iterated morphisms, the decreasing length conjecture: if \(h:\Sigma^*\to\Sigma^*\) is a morphism with \(| \Sigma |=n\), and \(w\) is a word with \(| w|>| h(w) | >| h^2(w) |>\cdots>| h^k(w)|\), then \(k\leq n\).