Chapter 7.4, Problem 12E

(a)

To determine

To prove: That for all values of $n\in \mathbb{N}$ , $a_n>a_{n+1}>b_{n+1}>b_n$ using induction.

(b)

To determine

To prove: That both sequences $a$ and $b$ converge if $a$ and $b$ are the sequences whose terms are $a_n$ and $b_n$ .

(c)

To determine

To prove: That ${\lim }_{n\to \infty }{a}_n={\lim }_{n\to \infty }{b}_n$ which is also called as arithmetic=geometric mean.