Marywood University Department of Mathematics Course Information for Math 430 Introduction to Real Analysis by Dr. J.

Karl Weierstrass (1815-1897)

An introduction to Real Analysis. This course is recommended for undergraduates who are considering going to graduate school in mathematics. It explores the foundations and underpinnings of the fundamental concepts of calculus along with the notion of proof in mathematics. Topics include completeness and supremum properties of the reals, sequences, convergence theorems, limits, continuous functions, differentiation, and an introduction to topology. The textbook is "Real Analysis", 3rd edition, by Bartle and Sherbert published by Wiley. The course grade is determined by three tests, five or six homework sets, and a final examination. Useful websites are listed at the end of this page. Prerequisite: Math 204 (Calculus IV); Recommended prerequisite:Math 311 (Differential Equations)

Sample Problems for Test #1

• True/False.
  • For any sets A and B in R, (A\B) union (B\A) = (A union B)\(A intersect B).
  • For any sets A and B in R that are subsets of the domain of the function f, f(A union B) = f(A) union f(B).
  • Every nonempty set of real numbers has a supremum in R.
  • Any subset of a countable set must be countable.
  • Any subset of an uncountable set must be uncountable.
  • The set of irrationals in the interval [0, 1] is uncountable.
• If f: A --> B is injective and if g: B --> C is injective, then prove that the composition g o f : A --> C is injective.
• If S:= { 1/n - 1/m : m, n members of N }, find inf S and sup S .
• Give an example of:
  • A function f : A --> B and a subset E of A such that f ^-1(f(E)) not = E.
  • A bounded set of real numbers that does not contain its supremum or its infimum.
  • A bounded function which does not attain its maximum value.
  • A nested set of closed intervals I_n, n in N, such that (intersection I_n )= {2}.
  • A bijection which demonstrates that the set of all odd integers is countable.
• Prove that Q is countable.
• Let J be the open interval J:= (a, b). For any x in J , show that there exists e > 0 such that V_e(x) is a subset of J.
• Let a > 0. Show that sup{a - 1/n : n in N } = a.
• Let S be a non-empty subset of R. Prove that u is an upper bound of S iff the conditions t in R and t > u imply that t is not an element of S.
• Let X be a non-empty subset of R and let f: X Æ R have a bounded range in R. If a in R, then show that:
  • sup{a + f(X)} = a + sup f(X)
  • inf{a + f(X)} = a + inf f(X)

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Sample Problems for Test #2

• True/False.
  • If lim(|x_n|) = |x| , then lim(x_n) = x .
  • A bounded sequence always has a convergent subsequence.
  • A monotone sequence always has a convergent subsequence.
  • If (x_kn ) is one subsequence of (x_n) such that lim (x_kn ) = x, then it must be true that lim (x_n) = x.
  • If (x_kn ) is one subsequence of (x_n) such that lim (x_kn ) not = x, then it is true that lim (x_n) not = x.
  • Every contractive sequence converges.
  • A Cauchy sequence must be monotone.
• Prove that lim [(n²-1)/(2n²+3)] = 1/2.
• Give an example of two divergent sequences (x_n) , (y_n) such that the product sequence (x_ny_n) converges.
• Prove that if lim (x_n) = x and if x > 0, then there exists a natural number M such that x_n > 0 for all n > M.
• Let x₁ > 2 and let x_n+1 = 4 - 4/x_n for n „ 2. Show that (x_n) is bounded and monotone. Find the limit.
• If (b_n) is a bounded sequence and lim (a_n) = 0, then show that lim (a_nb_n) = 0.
• Let (x_n) be a bounded sequence and let s:= sup{x_n}. Show that if s is not an element of {x_n}, then there exists a subsequence of (x_n) that converges to s.
• If (x_n) and (y_n) are Cauchy sequences, prove that (x_n + y_n) is a Cauchy sequence.

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Useful Websites

http://www.shu.edu/projects/reals/ (On-line course in real analysis. Contains key definitions and theorems.)

http://www.maa.org (Mathematical Association of America)

http://archives.math.utk.edu (Math Archives)

http://forum.swarthmore.edu (Math Forum)

Marywood University <|> Undergraduate School <|> Undergraduate Admissions

Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@ac.marywood.edu

Last update:September 5, 2001

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