2 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } = 1\) and
$$u _ { n + 1 } = - 1 + \sqrt { } \left( u _ { n } + 7 \right)$$
- Prove by induction that \(u _ { n } < 2\) for all \(n \geqslant 1\).
- Show that if \(u _ { n } = 2 - \varepsilon\), where \(\varepsilon\) is small, then
$$u _ { n + 1 } \approx 2 - \frac { 1 } { 6 } \varepsilon$$