91Ó°ÊÓ

Suppose the real numbers \(r\) and \(s\) satisfy \(0

Let \(x_{0}=1\), and for \(n \geq 0\) let \(x_{n+1}=\frac{x_{n}^{2}+2}{2 x_{n}}\). a. For any \(n \geq 0\), show that \(x_{n}\) is a rational number with \(1 \leq x_{n} \leq 2\). b. For any \(n \geq 0\), show that \(\left|x_{n}^{2}-2\right| \leq \frac{1}{4^{2^{2}-1}}\). c. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\sqrt{2}\).

Prove the following modification of the Principle of Mathematical Induction that allows us to use a basis for induction other than \(n=1\). Let \(n_{0}\) be an integer. Suppose for any integer \(n \geq n_{0}\) that \(S_{n}\) is a statement about \(n\). Suppose \(S_{n_{0}}\) is true and for any integer \(n \geq n_{0}\) the implication \(S_{n} \Longrightarrow S_{n+1}\) is true. Then \(S_{n}\) is true for all integers \(n \geq n_{0}\). (Suggestion: Apply the Principle of Mathematical Induction in its original form to the statement \(S_{n}^{t}=S_{n+m-1}\).)

Use the Principle of Mathematical Induction to prove the following principle, known as the Principle of Complete Mathematical Induction. Suppose \(S_{n}\) is a statement about the natural number \(n\). Suppose \(S_{1}\) is true and for any natural number \(n\) the implication \(S_{k}\) for all \(k \leq n \Longrightarrow S_{n+1}\) is true. Then \(S_{n}\) is true for all natural numbers.

Prove that the intersection of any collection of inductive subsets of \(\mathbb{R}\) is an inductive set.