Math 242: Principles of Analysis Fall 2016 Exam 1 Information . Department of Mathematics and Computer Science College of the Holy Cross

General Information

Important Definitions/Theorems/Axioms

• You should know precise statements of the following definitions and theorems:
  • Definition 1.5.2
  • Definition 1.5.3
  • Least Upper Bound Axiom
  • Theorem 1.5.13 (Archimedian Property)
  • Definition 2.1.4
  • Definition 2.2.2
  • Definition 2.3.1
  • Theorem 2.3.3 (Monotone Convergence Theorem)
  • Theorem 2.5.12 (Bolzano-Weierstrass Theorem)
• I will ask you to write a complete proof of one of the following theorems:
  • Theorem 1.5.16
  • Theorem 2.2.5, part (a)
  • Theorem 2.3.3 (Monotone Convergence Theorem)
• I will expect you to know how to use all of the definitions and theorems below to prove results similar to those on the homework assignments.

• I will not expect you to memorize the Axioms of the real numbers. However, I will ask you to use the Axioms (which would be given to you) to prove something (such as one part of Theorem 1.3.1).
• Triangle Inequality (1.3.4 (g))
• Upper and Lower Bounds for a set A (1.5.2)
• Least Upper Bound and Greatest Lower Bound for a set A (1.5.3)
• The Least Upper Bound Axiom
• The Archimedian Property (1.5.13)
• Density of Q (and I) in R. (1.5.14, 1.5.15 and 1.5.18)
• Existence of Square Roots (1.5.16)

• Principle of Mathematical Induction.

• Convergence of a Sequence (Definition 2.1.4).
• All of the Limit Theorems in Section 2.2.
• The Monotone Convergence Theorem
• Subsequences: Theorem 2.5.8 and the Bolzano Weierstrass Theorem.

Practice Questions