Every binary expansion represents a real number x:
x
= ±N.b_{1}b_{2}…,
b_{k}
∈ {0,1}.
This is the statement that every infinite series of the form
b_{1} 2^{−1} + b_{2} 2^{−2} + …, b_{k} ∈ {0,1},
converges.
•
Bolzano-Weierstraß Property: Every bounded sequence
{x_{k}} has an accumulation point: there is a real number
x such that every neighborhood of x contains x_{k} 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 {x_{k}} 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
F_{ϵ}
such that if j, k ∉ F_{ϵ}, then |x_{j} − x_{k}| < ϵ.
Useful Theorems about Continuous Functions
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
{z_{k}} of complex numbers has an accumulation point: there is a complex number
z such that every neighborhood of z contains z_{k} 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 points^{1}: if {z_{k}}
is a sequence of points in A, and lim_{k → ∞} z_{k} = 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
set^{2} 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:
^{1}A 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.
^{2}A closed and bounded set is a
compact set
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