91Ó°ÊÓ

Let \(X\) and \(Y\) be nonempty sets and let \(h: X \times Y \rightarrow \mathbb{R}\) have bounded range in \(\mathbb{R}\). Let \(F: X \rightarrow \mathbb{R}\) and \(G: Y \rightarrow \mathbb{R}\) be defined by $$ F(x):=\sup \\{h(x, y): y \in Y\\}, \quad G(y):=\sup \\{h(x, y): x \in X\\} $$ Establish the Principle of the Iterated Suprema: $$ \sup \\{h(x, y): x \in X, y \in Y\\}=\sup \\{F(x): x \in X\\}=\sup \\{G(y): y \in Y\\} $$ We sometimes express this in symbols by $$ \sup _{x, y} h(x, y)=\sup _{x} \sup _{y} h(x, y)=\sup _{y} \sup _{x} h(x, y) $$

(a) Give the first four digits in the binary representation of \(\frac{1}{3}\). (b) Give the complete binary representation of \(\frac{1}{3}\).

Show that a nonempty finite set \(S \subseteq \mathbb{R}\) contains its supremum. [Hint: Use Mathematical Induction and the preceding exercise.]

Determine and sketch the set of pairs \((x, y)\) in \(\mathbb{R} \times \mathbb{R}\) that satisfy: (a) \(|x|=|y|\), (b) \(|x|+|y|=1\), (c) \(|x y|=2\) (d) \(|x|-|y|=2\).

If \(y>0\), show that there exists \(n \in \mathbb{N}\) such that \(1 / 2^{n}

If \(0 \leq a

Determine and sketch the set of pairs \((x, y)\) in \(\mathbb{R} \times \mathbb{R}\) that satisfy: (a) \(|x| \leq|y|\), (b) \(|x|+|y| \leq 1\), (c) \(|x y| \leq 2\), (d) \(|x|-|y| \geq 2\).

Find all. real numbers \(x\) that satisfy the following inequalities. (a) \(x^{2}>3 x+4\), (b) \(1

Express \(\frac{1}{7}\) and \(\frac{2}{19}\) as periodic decimals.

Let \(a, b \in \mathbb{R}\), and suppose that for every \(\varepsilon>0\) we have \(a \leq b+\varepsilon\). Show that \(a \leq b\).