Midterm I Review for Math 421, Fall 2009

Technical terms you must know.

You must be able to give a careful, complete and accurate of these terms. You must also know examples of all of these terms (and their negations or combinations if that makes sense).
  1. a rational number
  2. an irrational number
  3. a set of real numbers which is bounded above or below
  4. an upper (lower) bound of a set of real numbers
  5. the least upper bound and greatest lower bound (sup and inf) of a set of real numbers
  6. the cardinality equivalence relation between sets, A~B
  7. a countable or uncountable set
  8. a sequence, the limit of a sequence, a convergent or divergent sequence
  9. the ε-neighborhood of a real number
  10. a bounded sequence
  11. an increasing or decreasing or monotone sequence
  12. a Cauchy sequence
  13. an infinite series, partial sums and convergence of such a series
  14. a subsequence
  15. the harmonic series, a geometric series
  16. the Cantor set
  17. an open set of real numbers

Theorems/Results/Axioms you must know.

This means you must be able to state them and to give examples of them being true and examples of how the conclusion is false if individual hypothesese are dropped.
  1. The Axiom of Completeness
  2. The Nested Interval Property
  3. The Archimedean Property
  4. The countability of Q and uncountability of R (the latter uses Cantor's Diagonalization Method)
  5. Results about the possible cardinalities of subsets of countable sets and of the power sets of given sets (the latter is Cantor's Theorem on Power Sets)
  6. The Algebraic and Order Limit Theorems
  7. The Monotone Convergence Theorem
  8. The Bolzano-Weierstrass Theorem
  9. the relationship between convergence and boundedness for sequences
  10. the relationship between a sequence being convergent and being Cauchy
  11. the construction of the Cantor set

Theorems you should be able to prove during an in-class midterm:

  1. The square root of a prime is irrational
  2. Q is countable and R uncountable
  3. Cantor's Theorem on Power Sets
  4. the MCT
  5. the Bolzano-Weierstrass Theorem