0.2. Real analysis 3

Let a1,a2,...,aN and b1,b2,...,bN be real numbers. Cauchy’s Inequality

asserts that

(0.4)

N

n=1

anbn ≤

N

n=1

an

2

1/2

N

n=1

bn

2

1/2

.

That this is so follows immediately from the algebraic identity

N

n=1

an

2

N

n=1

bn

2

−

N

n=1

anbn

2

=

1

2

N

m=1

N

n=1

(ambn −

anbm)2,

since a sum of squares of real numbers is nonnegative. Moreover, from the above

identity it is clear that equality holds in (0.4) if and only if the two sequences {an},

{bn} are proportional.

The Principle of Mathematical Induction is one of the axioms that define the

integers. It can be formulated in a number of (equivalent) ways.

(1) Weak induction: If S is a set of positive integers, if 1 ∈ S, and if n + 1 ∈ S

whenever n ∈ S, then S is the set of all positive integers.

(2) Strong induction: If S is a set of positive integers, if 1 ∈ S, and if k ∈ S for

all positive integers k n implies that n ∈ S, then S is the set of all positive

integers.

(3) Well ordering: If S is a set of positive integers, and S is non-empty, then S

contains a least member.

In all three cases we are inducting from 1, but of course one could instead induct

from 0 or any other convenient point.

The Binomial Theorem is treated in Appendix B. A catalogue of trigonometric

formulæ is provided in Appendix T, for convenience. The manner in which we

express cos nθ as a polynomial in cos θ is the subject of Appendix C.

0.2. Real analysis

It is not our purpose to summarize all of real analysis. We mention only specific

items that we need, and these are largely concerned with such issues as conditions

that ensure that (a) one can exchange two limiting operations; and (b) a sequence

that appears to tend to a limit does so.

Theorem 0.3. A bounded monotonic sequence of real numbers has a limit.

A sequence {xn} is said to be a Cauchy sequenceif

(0.5) lim

m→∞

n→∞

(xm − xn) = 0 .

Clearly any sequence that tends to a finite limit is a Cauchy sequence. What

is important is that the converse is also true:

Theorem 0.4. If {xn} is a Cauchy sequence, then limn→∞ xn exists and is finite.