Let \(E\) be a nonempty subset of an ordered set \(S\). Suppose \(\alpha\) is a lower bound of \(E\) and \(\beta\) is an upper bound of \(E\). Prove that \(\alpha \le \beta\).
Let \(A\) be a nonempty set of real numbers which is bounded below, and let \(-A\) be the set of all numbers \(-x\) with \(x \in A\). Prove that
\[\inf A = -\sup(-A).\]A nonempty subset of an ordered set is called bounded if it is bounded above and bounded below.
Show that a nonempty \(A \subseteq \R\) is bounded if and only if there exists \(M \ge 0\) such that \(\lvert x \rvert \le M\) for every \(x \in A\).
Prove that if \(A, B \subseteq \R\) are nonempty and bounded, then
\[A + B = \{a + b \;;\; a \in A,\, b \in B\} \quad\text{and}\quad A \cdot B = \{ab \;;\; a \in A,\, b \in B\}\]are both bounded.
Prove that if \(A \subseteq \R\) is nonempty and bounded then
\[\sup_{x \in A} \lvert x \rvert = \max\{\lvert \sup A \rvert, \lvert \inf A \rvert\}.\]Let \(A, B \subseteq \R\) be nonempty sets such that
\[a \le b \qquad \text{for every } (a,b) \in A \times B,\]and such that for every \(\varepsilon > 0\) there exists \((a,b) \in A \times B\) with \(\lvert a - b \rvert < \varepsilon\). Show that \(A\) is bounded above, that \(B\) is bounded below, and that \(\sup A = \inf B\).
Prove that if \(A, B \subseteq \R\) are nonempty and bounded above, then
\[\min\{\sup A, \sup B\} = \sup\{\min\{a,b\} \;;\; (a,b) \in A \times B\}.\]If \(r\) is rational, \(r \neq 0\), and \(x\) is irrational, prove that \(r + x\) and \(rx\) are irrational.
Prove that there is no rational number whose square is \(12\). You can prove this directly or use 1.
Prove that if \((a,b,c) \in \Q^3\) then
\[a + b \sqrt[3]{2} + c \sqrt[3]{4} = 0\]if and only if \(a = b = c = 0\). Hint: Try multiplying the equation by \(\theta = 2^{1/3}\), and solving a linear system of equations for $\theta$ and $\theta^2$.
Let
\[C = \{\sqrt{n} - \lfloor \sqrt{n} \rfloor \;;\; n \in \N\},\]where \(x \mapsto \lfloor x \rfloor\) is the floor function and \(\N\) is the set of whole numbers. Show that \(C\) is a bounded subset of \(\R\), and find with proof \(\inf C\) and \(\sup C\).
A nonempty subset of \(\R\) is called an interval if it has the property that any number lying between two numbers in the subset also belongs to the subset. Assume that \(B\) is a bounded interval and that \(A \subseteq \R\) is nonempty and satisfies
\[\lvert x - y \rvert < \sup\{u - v \;;\; u, v \in B,\, u \ge v\} \qquad \text{for every } x, y \in A.\]Show that \(A + B\), defined as in Problem 3, is an interval.