MAT 320 Introduction to Analysis Fall 1996

Review for Final Examination

Questions will involve the following topics:

1. (A and B-tracks)

• 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 prove that if the numbers L and L' both are the limit of the sequence a_1, a_2, a_3, . . . , then L = L'.

• 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].

2. ( A-track: everything; B-track bold-face material.)

• Understand the definition of continuity in terms of convergent sequences (p. 69); be able to use it to prove that, for example, the function f(x) = x^2 is continuous at x=1, and that the function f(x)=0 if x< 1, =2 if x > =1 is not. (Examples 1 and 2 on pp. 70,71).

• Understand the ``delta-epsilon'' definition of continuity and be able to prove that it is equivalent to the ``convergent sequences'' definition (Theorem 3.1.3).

• Be able to prove: a continuous function on a closed interval is bounded (Theorem 3.2.1). Understand why a continuous function on a closed interval takes on its maximum and minimum values, and all values in between [Theorems 3.2.2 and 3.2.3].

• Understand the definition of uniform continuity and be able to show, for example, that the function f(x) = x^2 is not uniformly continuous when considered as a function defined on the whole number line. Be able to prove that a continuous function on a closed interval is uniformly continuous (Theorem 3.2.5). Understand the definition of Lipschitz continuity (Problem 7 p.83).

• Understand the definition of Riemann integrability (p.86) for a bounded function on a finite interval. Be able to show that the function which is 1 on rationals and 0 on irrationals (Example 1 p.99) is not Riemann integrable. Be able to prove that a continuous function on a closed interval is Riemann integrable (Theorem 3.3.1).

• Understand the definition of a Riemann sum (p.87) and why for a continuous function on a closed interval arbitrary Riemann sums converge to the Riemann integral (Corollary 3.3.2). This justifies the use of left-hand and right-hand sums in computations.

• Understand how the properties of the Riemann integral given in Theorems 3.3.3, 3.3.4, 3.3.5, Corollary 3.3.6, Theorem 3.3.7 follow from the definition and Corollary 3.3.2. Understand how to use Theorem 3.3.5 (the ``triangle inequality for integrals") and be able to prove it.

• Understand how the triangle inequalities (regular and ``for integrals") are used in the estimate of the error in a Riemann sum approximation in terms of a bound on the derivative (Theorem 3.4.1). Be able to apply this theorem to estimate how fine a partition is needed to obtain a desired accuracy (Example 1 p.93).

• Understand the definition of improper integrals in terms of limits, as used in Examples 3,4,5 on pp.105,106. Understand the convergence of the integral of sinx/x (Example 6 p.106).

• Understand the definition of ``f is differentiable at x,'' and be able to prove that this implies continuity at x (Theorem 4.1.1). Be able to prove the product rule (Theorem 4.1.2b).

• Be able to prove that the derivative vanishes at a maximum, with the correct hypotheses (Theorem 4.2.1), as well as Rolle's Theorem (Theorem 4.2.2) and the Mean Value Theorem (Theorem 4.2.3).

• Be able to prove the version of the Fundamental Theorem given as Theorem 4.2.5.

• Understand what the Taylor polynomials are and how to compute them (p.134). Be able to quote and use Taylor's Theorem estimating the error in approximating a function by its nth Taylor polynomial (Theorem 4.3.1).

• Understand how Newton's method works. Understand the necessity for the condition that the initial guess be sufficiently close to the root in Theorem 4.4.1.

• Understand the definitions of pointwise convergence and uniform convergence for a sequence of functions defined on a set E. Understand why the sequence f_n= x^n on [0,1] converges but does not converge uniformly.

• Be able to prove that a uniform limit of continuous functions is continuous.

• Understand the ``sup norm'' on the set of functions defined on a set E, and that convergence in the sup norm is the same as uniform convergence. Understand what a Cauchy sequence of functions is (in the sup norm) and be able to apply Theorem 5.3.2: A Cauchy sequence of continuous functions converges uniformly to a limit which is a continuous function.