## Review for Midterm 1

References are to Reed, Fundamental Ideas of Analysis pre-publication version, Wiley 1996

There will be four questions, each worth 25 points. One question will be: Prove that if the numbers L and L' both are the limit of the sequence a_1, a_2, a_3, ... then L = L'.

The other questions will involve the following topics:

• Methods of proof: understand how to set up and carry through a proof by induction [e.g. Proposition 1.4.3, Problems 7, 9 on p.23]. Be able to carry out a proof by contradiction [e.g. Proposition 1.4.2]. Understand the words ``converse, contrapositive, counterexample.''

• Cardinality: understand how to show that 2 sets have the same number of elements [Section 1.3, first paragraph, Example 1, Proposition 1.3.3, etc.] Understand the proof that the set of rational numbers is countable, and that the set of reals between 0 and 1 is not [Theorems 1.3.4 and 1.3.5, Problem 8 on p.18].

• Convergence of sequences of real numbers: understand and be able to apply the definition of ``the sequence {a_n} converges to the limit L.'' Be able to calculate N from epsilon as in [Problems 2 and 3 on page 31].

• Be able to apply the limit theorems 2.2.3, 2.2.4, 2.2.5, 2.2.6 to calculate the limit of a sequence defined recursively [ Problem 1(a) on p.64, proof of Theorem 2.7.1, proof of Theorem 2.7.2 (assuming the limit exists, what is it?)]. Also [Problem 6 on page 37].

• Cauchy sequences: understand and be able to apply the definition [Definition, p.43, Example 1]. Understand the importance of the Completeness Axiom, and be able to show that the rationals are not complete [discussion on page 45, last paragraph].

• Every bounded monotone sequence of real numbers converges to a limit [Theorem 2.4.3]. Understand how this theorem is used in the proof of Theorem 2.7.1, for example, and in the proof of the least upper bound property, Theorem 2.5.1.

• Least upper bounds: understand how their existence is proved from the Completeness Axiom [Theorem 2.5.1]; know how to prove that least upper bounds are unique [ Problem 3 on p.52].

• The Bolzano-Weierstrass Theorem: understand what a subsequence is; understand that there are bounded sequences that do not converge to a limit, understand the proof that a bounded sequence must have a convergent subsequence [Theorem 2.6.2].

• Limit points: understand the definition (p.53), and that limit points of a sequence are the same as limits of subsequences [Proposition 2.6.1].