91Ó°ÊÓ

For each of the following statements, determine whether it is true or false and justify your answer. a. The set \(\mathbb{Z}\) of integers is dense in \(\mathbb{R}\). b. The set of positive real numbers is dense in \(\mathbb{R}\). c. The set \(\mathbb{Q} \backslash \mathbb{Z}\) of rational numbers that are not integers is dense in \(\mathbb{R}\).

Write out the Difference of Powers Formula explicitly for \(n=4\) and 5 .

For each of the following statements, determine whether it is true or false and justify your answer. a. Every nonempty set of real numbers that is bounded above has a largest member. b. If \(S\) is a nonempty set of positive real numbers, then \(0 \leq\) inf \(S\). c. If \(S\) is a set of real numbers that is bounded above and \(B\) is a nonempty subset of \(S,\) then \(\sup B \leq \sup S\)

Suppose that \(S\) is a nonempty set of integers that is bounded below. Show that \(S\) has a minimum. In particular, conclude that every nonempty set of natural numbers has a minimum.

Use the Principle of Mathematical Induction to prove that for a natural number \(n\), $$ \sum_{j=1}^{n} j^{2}=\frac{n(n+1)(2 n+1)}{6} $$

Prove that if \(n\) is a natural number greater than \(1,\) then \(n-1\) is also a natural number.

Show that the Archimedean Property is a consequence of the assertion that every interval \((a, b)\) contains a rational number.

a. Prove that the sum of a rational number and an irrational number must be irrational. b. Prove that the product of two nonzero numbers, one rational and one irrational, is irrational.

(Bernoulli's Inequality) Show that for a natural number \(n\) and a nonnegative number \(b\) $$ (1+b)^{n} \geq 1+n b $$

Use Proposition 1.2 to show that there is no rational number whose square equals \(2 / 9\).