4 The sequence \(\left\{ A _ { n } \right\}\) is given for all integers \(n \geqslant 0\) by \(A _ { n } = \frac { I _ { n + 2 } } { I _ { n } }\), where \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x d x\).
- Show that \(\left\{ A _ { n } \right\}\) increases monotonically.
- Show that \(\left\{ \mathrm { A } _ { \mathrm { n } } \right\}\) converges to a limit, \(A\), whose exact value should be stated.