T. Ratliff Home  |  Math Home  |   Wheaton Home

Major Terms and Results - Math 301 Real Analysis - Fall 2017

This page uses MathJax to display mathematical notation, so please let me know if any part isn't clear.

Be sure to check back, because there will certainly be some changes during the semester.

All numbers indicate sections from Understanding Analysis, Second Edition by Abbott.

\( \def\R{\mathbb{R}} \def\N{\mathbb{N}} \def\Q{\mathbb{Q}} \def\e{\epsilon} \def\dst{\displaystyle} \)

Chapter 1 The Real Numbers

Theorem 1.1.1. There is no rational number whose square is 2.

Theorem 1.2.6.
Two real numbers \( a\) and \( b\) are equal iff for every real number \( \e>0\) it follows that \( |a-b| <\e \).

Principle of Induction.
If \(S\subset \N\) such that

  1. \( S\) contains 1 and
  2. whenever \( S\) contains \( n\in\N \), it also contains \( n+1\),
then \( S = \N \).

Axiom of Completeness.
Every nonempty set of real numbers that is bounded above has a least upper bound.

Definition 1.3.1.
A set \( A\subset \R\) is bounded above if there exists a number \( b\in\R\) such that \( a\le b\) for all \( a\in A\). The number \(b\) is called an upper bound for \(A\).

Similarly, the set \(A\) is bounded below if there exists a lower bound \( l\in\R\) satisfying \( l\le a\) for every \( a\in A\).

Definition 1.3.2.
A real number \(s\) is the least upper bound, or supremum for a set \( A\subset \R\) if it meets the following two criteria:

  1. \( s\) is an upper bound for \( A\)
  2. if \( b\) is any upper bound for \( A\), then \( s \le b \)
The greatest lower bound, or infimum is defined similarly.

Lemma 1.3.8.
Suppose \(s\in\R\) is an upper bound for \(A\subset\R\). Then \(s=\sup A\) iff \(\forall \e>0\ \exists\ a\in A\ \ni s-\e < a\).

Theorem 1.4.1 (Nested Interval Property).
For each \(n\in \N\), assume that we have a closed interval \(I_n=[a_n,b_n]\) such that \[ I_1 \supset I_2 \supset I_3 \supset I_4 \supset \cdots\] Then \(\displaystyle \bigcap_{n=1}^{\infty} I_n \ne \emptyset \)

Theorem 1.4.2 (Archimedean Property).

  1. Given any \(x\in\R\), there exists \(n\in\N\) such that \(n>x\).
  2. Given any real number \(y>0\), there exists \(n\in\N\) such that \(\frac{1}{n} < y\).

Theorem 1.4.3 (Density of \( \Q\) in \(\R\)).
For every two real numbers \(a\) and \(b\) where \(a< b\), there exists a rational \(r\) satisfying \(a< r < b\).

Corollary 1.4.4.
Given any two real numbers \( a < b\), there exists an irrational number \( t\) satifying \( a < t < b\).

Theorem 1.4.5 .
There exists a real number \( \alpha\in\R\) satisfying \( \alpha^2 = 2\).

Definition 1.5.1.
A function \( f:A\to B\) is one-to-one if \( a_1 \ne a_2\) in \( A\) implies that \( f(a_1) \ne f(a_2)\) in \(B\). The function is onto if, given any \(b\in B\), it is possible to find an element \( a\in A\) for which \(f(a)=b\).

Definition 1.5.2.
A set \(A\) has the same cardinality as \(B\) if there exists \(f:A\to B\) that is one-one and onto. In this case we write \( A\sim B\).

Definition 1.5.5.
A set \(A\) is countable if \( \N \sim A\). An infinite set that is not countable is called an uncountable set.

Theorem 1.5.6.

  1. The set \(\Q\) is countable.
  2. The set \(\R\) is uncountable.

Theorem 1.5.7.
If \(A\subset B\) and \(B\) is countable, then \(A\) is either countable or finite.

Theorem 1.5.8.

  1. If \(A_1,A_2, \ldots,A_m\) are each countable sets, then the union \(A_1\cup A_2\cup\cdots\cup A_m\) is countable.
  2. If \(A_n\) is a countable set for each \(n\in\N\), then \(\dst\cup_{n=1}^\infty A_n\) is countable.

Theorem 1.6.1.
The open interval \( (0,1)\subset\R \) is uncountable.

Theorem 1.6.2 (Cantor's Theorem).
Given any set \( A\), there does not exist a function \( f:A\to P(A)\) that is onto.

Chapter 2 Sequences and Series

Definition 2.2.3.
A sequence \( (a_n) \) converges to a real number \( a\) if for every \( \e > 0 \) there exists \( N\in\N \) such that \( n\ge N\) implies \( | a_n - a | < \e\).

Definition 2.2.4.
Given a real number \( a\in \R\) and a positive number \( \e>0\), the set \[ V_{\e}(a) = \{ x\in\R \mid |x-a| < \e \} \] is called the \( \e\)-neighborhood of \( a\) .

Definition 2.2.3B (Convergence of a Sequence: Topological Version).
A sequence \( (a_n)\) converges to a if, given any \(\epsilon\)-neighborhood \( V_{\e}(a)\) of a, there exists a point in the sequence after which all of the terms are in \( V_{\e}(a)\). In other words, every \(\epsilon\)-neighborhood contains all but a finite number of the terms of \( (a_n)\).

Understanding Analysis

Maintained by: ratliff_thomas@wheatoncollege.edu