The sequence of positive numbers \(u_{1}, u_{2}, u_{3}, \ldots\) is such that \(u_{1}\lt 3\) and, for \(n \geqslant 1\),
\(u_{n+1}=\frac{4 u_{n}+9}{u_{n}+4} .\)
(i) By considering \(3-u_{n+1}\), or otherwise, prove by mathematical induction that \(u_{n}\lt 3\) for all positive integers \(n\).

(ii) Show that \(u_{n+1}\gt u_{n}\) for \(n \geqslant 1\).