MATH 8, SECTION 1, WEEK 2  RECITATION NOTES
TA: PADRAIC BARTLETT
Abstract.
These are the notes from Wednesday, Oct.
6th’s lecture, where
we showed that there are infinite sets with different cardinalities.
1.
Random Question
Question 1.1.
Define
α
(
n
)
as the function that takes in an integer and returns the
total number of prime divisors of
n
: for example
α
(9) = 2
, α
(36) = 4
,
and
α
(7) = 1
.
Find
lim
n
→∞
α
(
n
)
n
.
2.
Introduction
The goal of this lecture is to prove the following:
Theorem 2.1.
The sets
N
and
R
have different cardinalities.
In our last lecture, we introduced the idea of cardinality, which allows us to make
sense of the idea of having “different” sizes of infinity; the last tool we need here
before we begin our proof, then, is to define the sets we are working with.
The
natural numbers are (hopefully, by this point) something we understand quite well:
the reals, however, we have not yet given a rigorous footing.
Let’s change that.
3.
Defining the Real Numbers
In
Q
, there are sequences
{
a
n
}
∞
n
=1
of rational numbers that have the three fol
lowing properties:
•
the terms
a
n
are increasing,
•
there is an upper bound for the terms
a
n
, but
•
this sequence of
a
n
’s doesn’t converge to a rational number.
Specifically, consider the following example:
•
a
0
= 1
•
a
1
= 1
.
4
•
a
2
= 1
.
41
•
a
n
= the first
n
decimal places of
√
2; i.e.
a
n
=
b
√
2
·
10
n
c
10
n
Proposition 3.1.
The sequence
{
a
n
}
∞
n
=1
defined above converges to
√
2
, an irra
tional number.
1
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TA: PADRAIC BARTLETT
Proof.
Before we begin our proof, we should probably say just what convergence
*is:*
Definition 3.2.
A sequence
{
a
n
}
∞
n
=1
converges to some value
λ
if, for any distance
, the
a
n
’s are eventually within
of
λ
. To put it more formally, lim
n
→∞
a
n
=
λ
iff for any distance
, there is some cutoff point
N
such that for any
n
greater than
this cutoff point,
a
n
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 Fall '09
 Calculus, Real Numbers, Sets, lim, TA, Decimal, upper bound

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