3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } > 4\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { u _ { n } ^ { 2 } + u _ { n } + 12 } { 2 u _ { n } }$$

1. By considering \(\mathrm { u } _ { \mathrm { n } + 1 } - 4\), or otherwise, prove by mathematical induction that \(\mathrm { u } _ { \mathrm { n } } > 4\) for all positive integers \(n\).
2. Show that \(u _ { n + 1 } < u _ { n }\) for \(n \geqslant 1\).