MathDB

The sequences

a_0, a_1, \ldots

and

b_0, b_1, \ldots

are defined for n \equal{} 0, 1, 2, \ldots by the equalities a_0 \equal{} \frac {\sqrt {2}}{2}, a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}} and b_0 \equal{} 1, b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n} Prove the inequalities for every n \equal{} 0, 1, 2, \ldots 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n.

Problem Statement