A total of \(n\) independent trials have been performed. The probability of the occurrence of event \(A\) in the \(i\) th trial is \(p_{i} ; P_{n}(m)\) is the probability of the \(m\)-fold occurrence of event \(A\) in \(n\) trials. Prove that

\[\begin{equation*} \frac{P_{n}(1)}{P_{n}(0)} \geqslant \frac{P_{n}(2)}{P_{n}(1)} \geqslant \ldots \geqslant \frac{P_{n}(n)}{P_{n}(n-1)} \tag{a} \end{equation*}\]

(b) \(P_{n}(m)\) first increases and then decreases (if \(P_{n}(0)\) or \(P_{n}(n)\) are not themselves maximal).