MathDB

Let

a_{0}=\sqrt{2}, b_{0}=2,a_{n+1}=\sqrt{2-\sqrt{4-a_{n}^{2}}},b_{n+1}=\frac{2b_{n}}{2+\sqrt{4+b_{n}^{2}}}

. a) Prove that the sequences

(a_{n})

and

(b_{n})

are decreasing and converge to

0

. b) Prove that the sequence

(2^{n}a_{n})

is increasing, the sequence

(2^{n}b_{n})

is decreasing and both converge to the same limit. c) Prove that there exists a positive constant

C

such that for all

n

the following inequality holds:

0 <b_{n}-a_{n} <\frac{C}{8^{n}}

Problem Statement