Lemma: Any Cauchy sequence whose range is finite, is convergent
Bolzano - Weierstrass Theorem: Every bounded infinite set has an
of real
accumulation point
numbers
Corollary: every Cauchy sequence converges
Heine Borel Theorem: A set is compact iff it is closed and bounded
Theorem: K is a subset of R, K is sequentially compact iff it is closed
and bounded
Closed and bounded = sequentially compact = compact
Proof: Suppose K is sequentially compact
First show
Theorem: Every bounded sequenced has a convergent subsequence
Proof:
Theorems about uniformly continuous functions
Similar to below, i.e. using "multiplication trick"
Topology of the real line
most sets are neither open nor closed, open and closed are NOT
L'Hopital's Rule:
Inverse Function Theorem:
Essentially: if the derivative is not equal to zero, then the inverse exists
Theorem:
f is one to one (theorem 4.9) and therefore the inverse function exists
Range is an interval because of intermediate value th
Riemann Integral: Definition
Let Q be a refinement of P obtained by adjoining K points then
Similarly for upper sums:
Theorem: Suppose f is a monotone function on [a,b] then f is
Riemann Integrable
(Note: it has at most countably many discontinuities)
Pro
Algebra of continuous functions:
If you add or multiply two continuous functions, the
result is continuous (by algebra of limits)
Want to show sin(x) is continuous
is uniformly continuous (on E) if
Criterion for non uniform continuity
Criterion for lack o
If f and g are continuous, then f/g is continuous where g =/= 0
Compact set:
Definition:
Lemma: Suppose K is compact, then K is closed (it contains all its
accumulation points) and bounded
So x_0 is not an accumulation point of K
Theorem 3.8: continuous o
Office hours: noonish until class and after class
Chapter 0
Sets: Capital letters for the set, lowercase letters for members
Power set
labeling:
indexed family of sets, index set is natural numbers
Cartesian product
A relation R between the sets A and B i
Definition: a sequence (also true for functions) is said to be bounded
if its range is bounded
Theorem: every Cauchy sequence is bounded
Correlary: every
convergent sequence is
bounded
Inverse:
Definition: An epsilon neighborhood of
(One dimensional disk
Real numbers aren't countable
Image and pre-image:
then g is an extension of f
Triangle Inequality
archemdian principle?
quiz on definitions on Friday, link on webpage for definitions: chapter
0: 20 definitions: know 7-20
Chapter 1:
Definition: Sequences:
DeMorgan's Laws for Sets on Real Numbers (only used in this class
for sets on the real line)
Relations on a collection of sets
Definition: We say a set S is
countable if there exists an
injection
Countable sets:
Any finite set,
Note:
1) if a set is counta
Criterion for nonconvergence
Applications:
Exam: chapter 0 and 1
Problems, proofs
Chapter 2: theorem 2.1, definition of limit of a function, criterion for
nonconvergence (2.1 and 2.2)
Algebra of limits
Application of math induction
Definition: given a sequence
Theorem 1.14: a sequence converges iff each of its subsequences
converges, and if they do converge, they all converge to the same limit
Proof:
Suppose all subsequences converge, since a sequence i
Definitions for tomorrow: ch1 : definitions: 1-
Theorem (Algebra of Limits)
Now use proof of 2
Monotone sequences
A sequence that is increasing or decreasing is called a monotone
sequence
Theorem: Every bounded monotone sequence converges
Theorem: suppose
First note that all of these sequences
must have the same limit
Suppose we have two special sequences:
Let L denote the common limit of all images of special sequences
Exam 1
If there are two acc pts, you can find a subsequence that
converges to each acc pt, but every subsequence must
converge to the same limit
If it converges, it is Cauchy.
Definition:
If you have an increasing function:
The limit from the right side (
Leibniz's Rule:
Change of Variables Theorem: Let J be an interval
Note: this is also know as the substitution method for integration
Mean Value Theorem for Integrals: