Homework 4: Functions, Proofs of Quantified Statements

Instructions. Assignments should be stapled and written clearly and legibly.

1. Section 2.3, #5(a)(b), 7(b)(c)(e), 8.
2. Section 1.2, #8, 9(d), 10(c), 11(f).
3. Section 1.4, #11.
4. Prove that for every integer \(b\), there exists a positive integer \(a\) such that \(|a-|b||\leq 1\).
5. Prove that for every positive real number \(e\), there exists a positive real number \(d\) such that if \(x\) is a real number with \(|x| \lt d\), then \(2|x|\lt e\).
6. Prove that for every positive real number \(\epsilon\), there exists a natural number \(N\) such that if \(n \gt N\), then \(\frac{1}{n^2+1} \lt \epsilon\).