A sequence (left(a_{n} ight)) is defined by [ a_{1}=1 text { and } a_{n+1}=frac{a_{n}^{2}+2}{2 a_{n}} forall n geq 1 ] (i) Prove that (left(a_{n} ight))

A sequence \(\left(a_{n}\right)\) is defined by

\[ a_{1}=1 \text { and } a_{n+1}=\frac{a_{n}^{2}+2}{2 a_{n}} \forall n \geq 1 \]

(i) Prove that \(\left(a_{n}\right)\) is a bounded sequence, and decreases for \(n \geq 2\).

(ii) Show that the limit of \(\left(a_{n}\right)\) is \(\sqrt{2}\).