Derangements in Sylow Subgroups of Symmetric Groups

Let \(H_n < S_n\), where \(H_n\) is a Sylow \(p\)-subgroup of \(S_n\), the symmetric group on \(n\) letters. Let \(A_n\) denote the number of derangements in \(H_n\), and \(f_n = \frac{h_n}{|H_n|}\). We will show that the sequence \(\{f_n\}_{n=1}^{\infty}\) is dense in the unit interval, but is Cesàro convergent to \(0\).