Let $a$ and $b$ be positive numbers with $a>b$. Let $a_1$ be their arithmetic mean and $b_1$ their geometric mean:
$$
a_1=\frac{a+b}{2} \quad b_1=\sqrt{a b}
$$
Repeat this process so that, in general,
$$
a_{n+1}=\frac{a_n+b_n}{2} \quad b_{n+1}=\sqrt{a_n b_n}
$$
(a) Use mathematical induction to show that
$$
a_n>a_{n+1}>b_{n+1}>b_n
$$
(b) Deduce that both $\left\{a_n\right\}$ and $\left\{b_n\right\}$ are convergent.
(c) Show that $\lim _{n \rightarrow \infty} a_n=\lim _{n \rightarrow \infty} b_{\text {a }}$. Gauss called the common value of these limits the arithmetic-geometric mean of the numbers $a$ and $b$.