4 0. Background

Note that a sequence of rational numbers tending to

√

2 is a Cauchy sequence,√

but does not not have a limit within the system of rational numbers, because 2

is irrational. In a set-theoretic sense, the real numbers are constructed by filling in

the holes found among the rational numbers. Because all Cauchy sequences have a

limit, we say that the real numbers are complete.

A function f is continuous at a if limx→a f(x) = f(a). That is, for every ε 0

there is a δ 0 such that |f(x) − f(a)| ε if |x − a| δ. Here the choice of δ

depends on both ε and a. However, if in a domain D we have |f(x) − f(y)| ε

whenever x ∈ D, y ∈ D and |x−y| δ, then we say that f is uniformly continuous

on D.

Theorem 0.5. If a real-valued function f(x) is continuous on a closed bounded

interval [a, b], then it is uniformly continuous on that interval, and attains its max-

imum and minimum values.

The same theorem also holds for real-valued continuous functions defined on

closed bounded sets in the plane

R2.

Theorem 0.6. (The Squeeze Theorem) Suppose that δ 0, that f−, f, and f+ are

functions such that f−(x) ≤ f(x) ≤ f+(x) for a−δ x a+δ. If limx→a f−(x) = c

and limx→a f+(x) = c, then limx→a f(x) = c.

Theorem 0.7. (Rolle’s Theorem) If f(x) is a continuous real-valued function on

the interval a ≤ x ≤ b with f(x) differentiable for a x b, and if f(a) = f(b),

then there exists a ξ ∈ (a, b) such that f (ξ) = 0.

Theorem 0.8. (The Mean Value Theorem of Differential Calculus) If f(x) is a

continuous real-valued function on the interval a ≤ x ≤ b and f(x) is differentiable

for a x b, then there exists a ξ ∈ (a, b) such that

f (ξ) =

f(b) − f(a)

b − a

.

Theorem 0.9. If

∑∞

n=1

|an| ∞, then the sum

∑∞

n=1

an converges.

Theorem 0.10. Suppose that amn ≥ 0 for all m and n. We form two sums:

(0.6)

∞

m=1

∞

n=1

amn ,

∞

n=1

∞

m=1

amn .

If either of these sums is finite, then the other one is also finite, and they are equal.

Theorem 0.11. If |amn| ≤ Amn for all m and n, and if

∞

m=1

∞

n=1

Amn ∞,

then the sums (0.6) converge and are equal.

Theorem 0.12. Let f(z) =

∑∞

k=0

akzk be a power series, and let R be defined by

the relation

1

R

= lim sup

n→∞

|an|1/n

.