Let $a , b > 0$.\\
(a) Prove that $\lim _ { n \rightarrow \infty } \left( a ^ { n } + b ^ { n } \right) ^ { 1 / n } = \max \{ a , b \}$.\\
(b) Define a sequence by $x _ { 1 } = a , x _ { 2 } = b$ and $x _ { n } = \frac { 1 } { 2 } \left( x _ { n - 1 } + x _ { n - 2 } \right)$ for $n > 2$. Show that $\left\{ x _ { n } \right\}$ is a convergent sequence.