Exercise 7, § 8

General Topology, IV

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

Let $(\varepsilon_n)_{n \geq 0}$ be a sequence of numbers each of which is 1 or -1. Show 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 is 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 each real number $x$ , such that $-2 \leq x \leq 2$ , there is a sequence $(\varepsilon_n)$ such that $x$ is equal to the limit of the corresponding sequence $(x_n)$ . For what values of $x$ is the sequence $(\varepsilon_n)$ unique? For what values of $x$ is it periodic? (see Exercise 3). \*

Solution

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