Exercise 7, § 8

General Topology, IV, p. 62

Bourbaki marks this one as harder than the rest, it is set in small type as supplementary.

Soit $(\varepsilon_n)_{n \geq 0}$ a sequence of numbers equal to $1$ or to $-1$ ; now prove that the real number

x_n = \varepsilon_0 \sqrt{2 + \varepsilon_1 \sqrt{2 + \varepsilon_2 \sqrt{2 + \cdots + \varepsilon_n \sqrt{2}}}}

is defined, and equal to

2 \sin \left( \frac{\pi}{4} \sum_{k=0}^n \frac{\varepsilon_0 \varepsilon_1 \ldots \varepsilon_k}{2^k} \right).

Deduce that $\lim_{n \to \infty} x_n$ exists and that for every real number $x$ such that $-2 \leq x \leq 2$ , there exists a sequence $(\varepsilon_n)$ such that $x$ is equal to the limit of the corresponding sequence $(x_n)$ ; for which values of $x$ is the sequence $(\varepsilon_n)$ unique? For which values of $x$ is it periodic (see IV, p. 62, exerc. 3)?*

Solution

No solution has been written yet.