Analysis Fact Sheet

Completeness of Real Numbers

The following are equivalent statements of the completeness of the real numbers.

Every decimal expansion represents a real number x:
 x
 = ±N.d1d2…,
 dk
 ∈ {0,1, ... ,9}.
This is the statement that every infinite series of the form
 d1 10−1 + d2 10−2 + …,     dk ∈ {0,1, ... ,9},
converges.

Every binary expansion represents a real number x:
 x
 = ±N.b1b2…,
 bk
 ∈ {0,1}.
This is the statement that every infinite series of the form
 b1 2−1 + b2 2−2 + …,     bk ∈ {0,1},
converges.

Bolzano-Weierstraß Property: Every bounded sequence {xk} has an accumulation point: there is a real number x such that every neighborhood of x contains xk for infinitely many k.
Informally we shall say that infinitely many terms of the sequence are close to x. An accumulation point of the sequence is also called a limit point of the sequence.

Every bounded sequence has a convergent subsequence.

Bounded increasing sequences have limits.

Every nonempty set of real numbers which is bounded above has a least upper bound.

Every Cauchy sequence has a limit.
In the language of Knopp: A sequence {xk} is a Cauchy sequence if almost all the terms are close together in the precise sense:
For every ϵ > 0, there is a a finite set Fϵ such that if j, k ∉ Fϵ, then |xj − xk| < ϵ.

The following results about continuous functions are actually equivalent to any of the above characterizations of the completeness of real numbers. Fairly straightforward proofs can be done by using the Bolzano-Weierstraß property and a proof by contradiction.

If f(x) is continuous on a closed interval [a,b], then f(x) is bounded on [a,b].

If f(x) is continuous on a closed interval [a,b], then f(x) is uniformly continuous on [a,b].
Uniformly continuity on [a,b] means that there a function ϕ(∆x) which is o(1) as ∆x →0 such that
 |f(x+ ∆x) − f(x) | ≤ ϕ(∆x)
for all x, x + ∆x ∈ [a,b].

If f(x) is continuous on a closed interval [a,b], then f(x) assumes its maximum value at some point in [a,b].

Completeness of Complex Numbers

The above statements have consequences for complex numbers:
The real and imaginary parts of complex numbers have decimal [binary] expansions.

Bolzano-Weierstraß Property: Every bounded sequence {zk} of complex numbers has an accumulation point: there is a complex number z such that every neighborhood of z contains zk for infinitely many k.
Informally we shall say that infinitely many terms of the sequence are close to z. An accumulation point of the sequence is also called a limit point of the sequence.

Every bounded sequence of complex numbers has a convergent subsequence.

Every Cauchy sequence of complex numbers has a limit.

Useful Theorems about Continuous Functions of a Complex Variable

The statements for complex numbers require the concept of closed set.

A set A of complex numbers is closed if it contains all its accumulation points1: if {zk} is a sequence of points in A, and limk → ∞ zk = z, then z ∈ A.

Fairly straightforward proofs of the following can be done by using a proof by contradiction and the Bolzano-Weierstraß property.

If f(z) is continuous on a closed and bounded set2 A, then f(z) is bounded on A.

If f(z) is continuous on a closed and bounded set A, then f(z) is uniformly continuous on A.
Uniformly continuity on A means that there a function ϕ(∆z) which is o(1) as ∆z → 0 such that
 |f(z+ ∆z) − f(z) | ≤ ϕ(∆z)
for all z, z + ∆z ∈ A.

If f(z) is continuous on a closed and bounded set A, then |f(z)| assumes its maximum value at some point in A.

### Footnotes:

1A peculiar viewpoint of closed would be: A set A of complex numbers is closed if it has the Bolzano-Weierstraß property when considered as an entity in itself.
2A closed and bounded set is a compact set

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On 16 Nov 2013, 20:55.