Math 341 Lecture #2
1.3-1.4: Completeness of R
1.3 Axiom of Completeness. We have seen that 2 is a gap in Q.
We think of Q as a subset of R and that R has no gaps.
This assumption about R is known as the Axiom of Completeness: Every nonempty
set of real n
Math 341 Lecture #3
1.4-1.5: Completeness of R, Cantors Theorem
How does Q sit inside R?
Theorem 1.4.3 (Density of Q in R). For every two real numbers a and b with
a < b, there exists an r Q such that a < r < b.
Proof. To keep things simple, assume that 0
Math 341 Lecture #4
2.1,2.2: Rearrangements, Limits
2.1 Rearrangements of Innite Series. For nite sums we have the commutative
and associative properties holding, but what about innite sums?
Example. For positive integers i and j, consider the numbers
2ij
Math 341 Lecture #5
2.3: The Algebraic and Order Limit Theorems
The point of having the logically tight denition of convergence of a sequence is so that
we can prove theorems about convergent sequences, not just rely on good guesses.
In a homework problem
Math 341 Lecture #6
2.4: The Monotone Convergence Theorem
Not all bounded sequences, like (1)n , converge, but if we knew the bounded sequence
was monotone, then this would change.
Denition 2.4.1. A sequence (an ) is increasing if an an+1 for all n N and
Math 341 Lecture #7
2.5: The Bolzano-Weierstrass Theorem
Denition 2.5.1. Let (an ) be a sequence of real numbers, and let n1 < n2 < n3 <
be a strictly increasing sequence of natural numbers. Then the sequence
an1 , an2 , an3 , . . . ,
is called a subsequ
Math 341 Lecture #8
2.6: The Cauchy Criterion
There is way to describe a convergence sequence without an explicit reference to its limit.
Denition 2.6.1. A real sequence (an ) is called Cauchy if, for every > 0 there exists
N N such that whenever m, n N w