Mark's Elements, I
Mark Hopkins
Mark's Elements
A Euclidean space is a complete metric space with certain special
properties relating to connectedness, the Pythagorean theorem and so on.
In the following axiomitization, I've factored out these extra properties
providing a classical treatment of Euclidean Geometry.
In addition, I've introduced some basic equivalence results for metric
spaces and inner product spaces which lead to simplified definitions for
both, and provide characterisations of each of these as well as Banach
spaces. The most interesting aspect of these developments is that a
constructive equivalent to the concepts of Cauchy sequences and their limits
can be provided from which the classical epsilon-delta type argument and
the limit concept is nearly absent! Indeed, the alternative concept used:
the Zeno Sequence is something that could have been easily discovered (as
it was) and used in ancient times. So I'm really not exaggerating too much
in saying that this a classical treatment of the subject.
Contents:
(1) The Axioms
(2) Lines, Betweenness, and the Map r |-> [A, r, B]
(3) The Pythagorean Theorem
(4) Representation of Inner Product Spaces
(5) Dimension
(6) The Equivalence of Zeno Sequences and Cauchy Sequences
(1) The Axioms
The basic concepts are the POINT and DISTANCE. The distance from a
point A to a point B is denoted AB, as is ordinarily done in geometry
texts. The following axioms are assumed:
AXIOM 1: AA = 0
AXIOM 2: AB > 0 unless A = B
AXIOM 3: Symmetry: AB = BA
AXIOM 4: The Triangle Inequality: AB + BC >= AC
AXIOM 5: Infinite Divisibility
Any distance can be divided by any number:
Given any two points A, B and any number N = 2, 3, 4, ...
there is a point C such that
AC + BC = AB = N AC
AXIOM 6: Infinite Extensibility
Any distance can be multiplied by any number:
Given any two points A, B and any number M = 2, 3, 4, ...
there is a point C such that
AB + BC = AC = M AB
AXIOM 7: If A, B and C are three points such that AB + BC = AC, then
for any point D:
AB CD^2 - AC BD^2 + BC AD^2 = AB AC BC
AXIOM 8: Zeno's Counter-Axiom
Let A1, A2, A3, ... be a sequence of points such that
An A_{n+1} <= 1/2 An A_{n-1} for n > 1
Then there is a point A such that:
An A <= 2 An A_{n+1}
AXIOM 9: There are 4 points, all equally distant from one another.
AXIOM 10: No 5 points are all equally distant from one another.
AXIOMS 1-4 give us the definition for a metric space. With the addition
of AXIOM 8, one has the equivalent of a Cauchy-complete metric space. There
is no loss of generality in stating it in this form.
AXIOM 9 and AXIOM 10 relate to dimensionality and make essential
use of the Euclidean nature of the underlying geometry (i.e., AXIOM 7) in
defining dimensionality in terms of the existence of equilateral simplices.
This is probably the simplest way to state the property of dimensionality
in terms of distances, when given a Euclidean metric.
(2) Lines, Betweenness and the Map r |-> [A, r, B]
The notion of betweenness, which is implictly used in AXIOMS 5-7 is
captured by the definition
C is between A and B <==> AC + CB = AB
We can also define lines and line segments in the appropriate fashion.
To show that lines are in one to one correspondence with the real numbers
and segments in one to one correspondence with the interval [0, 1] will
then require a few technical results, which is what the initial part of
this exposition is concerned with.
THEOREM 1: Let A, B and C be points for which AC = |r| AB, BC = |1 - r| AB.
Then for any point D,
CD^2 = (1 - r) AD^2 + r BD^2 - r(1 - r) AB^2.
Proof:
There are three cases that need to be distinguished:
If r <= 0: AB + AC = BC. Therefore, using AXIOM 7, we get:
AB CD^2 - BC AD^2 + AC BD^2 = AB AC BC
AB CD^2 = AB (|1 - r| AD^2 - |r| BD^2 + |r(1 - r)| AB^2)
= AB ((1 - r) AD^2 + r BD^2 - r(1 - r) AB^2)
If 0 <= r <= 1: AC + BC = AB. This time, using AXIOM 7, we get:
AC BD^2 - AB CD^2 + BC AD^2 = AB AC BC
AB CD^2 = AB (|1 - r| AD^2 + |r| BD^2 - |r(1 - r)| AB^2)
= AB ((1 - r) AD^2 + r BD^2 - r(1 - r) AB^2)
If 1 <= r: AB + BC = AC. Therefore, by AXIOM 7, we get:
AB CD^2 - AC BD^2 + BC AD^2 = AB AC BC
AB CD^2 = AB (-|1 - r| AD^2 + |r| BD^2 + |r(1 - r)| AB^2)
= AB ((1 - r) AD^2 + r BD^2 - r(1 - r) AB^2)
The case AB = 0 is easily dispensed with, since then one would have
AC = |r| AB = 0, and therefore A = B = C by AXIOMS 1 and 2. From this
it would follow that:
(1 - r) AD^2 + r BD^2 - r(1 - r) AB^2
= (1 - r) CD^2 + r CD^2 - r(1 - r) 0^2 = CD^2
If AB > 0 then we can cancel the factor AB in all three cases above and
arrive at the same conclusion.
THEOREM 2: Given two points A and B and number r, there is at most one
point C such that AC = |r| AB, BC = |1 - r| AB.
Proof:
Suppose C and D are two points such that BC = BD = |1 - r| AB, and
AC = AD = |r| AB. Then by THEOREM 1, it follows that:
CD^2 = (1 - r) AD^2 + r BD^2 - r(1 - r) AB^2
= ( (1 - r) r^2 + r (1 - r)^2 - r(1 - r) ) AB^2
= 0
Therefore CD = 0, or C = D.
DEFINITION 1: If A, B and C are three points such that
AC = |r| AB, BC = |1 - r| AB
then write
C = [A, r, B]
By THEOREM 2, this establishes a partial one-to-one correspondence
between the real numbers and points on the interval AB, when A and B are
distinct points. Using this definition, THEOREM 1 can be restated in
the following form:
COROLLARY 3: If C = [A, r, B], then
CD^2 = (1 - r) AD^2 + r BD^2 - r(1 - r) AB^2
We list a few more basic properties of this partial function below.
THEOREM 4: If C = [A, r, B] and D = [A, s, B] then CD = |r - s| AB.
Proof:
Let C and D be as given. Then
CD^2 = (1 - r) AD^2 + r BD^2 - r(1 - r) AB^2
Since AD = |s| AB and BD = |1 - s| AB, then it follows that
CD^2 = ( (1 - r) s^2 + r (1 - s)^2 - r(1 - r) ) AB^2
= (r - s)^2 AB^2
or CD = |r - s| AB.
THEOREM 5: If C = [A, r, B] then [B, 1 - r, A] exists and is equal to C.
Proof:
Let C = [A, r, B]. Then AC = |r| AB and BC = |1 - r| AB. But these
conditions also uniquely define [B, 1 - r, A], since AC = |1 - (1 - r)| AB.
THEOREM 6: If C = [A, r, B] and r is not 0, then B = [A, 1/r, C].
Proof:
Let C = [A, r, B]. Then AC = |r| AB and BC = |1 - r| AB. From this
we get
AB = |1/r| AC, BC = |(1-r)/r| AC = |1 - 1/r| AC
which implies that B = [A, 1/r, C].
With the following results, we can identify the points given by AXIOMS 5
and 6, and then show that [A, r, B] is defined for all rational r.
THEOREM 7: If A, B and C are points for which AC + BC = AB = N AC, for any
number N > 1, then C = [A, 1/N, B].
Proof:
If points A, B and C are as stated above, then:
AC = 1/N AB = |1/N| AB, BC = AB - AC = (1 - 1/N) AB = |1 - 1/N| AB
Therefore, C = [A, 1/N, B].
THEOREM 8: If A, B and C are points for which AB + BC = AC = M AB, for any
number M > 1, then C = [A, M, B].
Proof:
Given the conditions stated above, one has:
AC = M AB = |M| AB, BC = AC - AB = (M - 1) AB = |1 - M| AB
Therefore, C = [A, M, B].
THEOREM 9: [A, r, B] is defined for all rational numbers r.
Proof:
There is no loss of generality in assuming that r > 0 since otherwise we
could first establish the existence of [B, 1 - r, A] and the use THEOREM 5
to identify this point as [A, r, B].
Assume that r > 0 and is given as r = M/N, with M, N > 1 (any fraction
can be expressed with both numerator and denominator larger than 1, e.g.
5/1 = 10/2). Let D = [A, 1/N, B] and C = [A, M, D]. Both points are
guaranteed to exist by AXIOMS 5 and 6, in conjunction with THEOREMS 7 and 8.
Then we have:
AC = M AD = M/N AB
B = [A, N, D] by THEOREM 6
BC = |N - M| AD by THEOREM 4
= |(N - M)/N| AB
= |1 - M/N| AB
Therefore, C = [A, M/N, B] = [A, r, B].
THEOREM 10: [A, r, B] exists for all real numbers r.
Proof:
Without loss of generality, we can restrict our attention to irrational
numbers r > 0, since we have THEOREMS 5 and 9 to handle the other cases.
First, we observe that every positive real number has a Zeno sequence of
rational numbers which converges to it. For example, consider the number
r = 1011.1010100...
written out in binary. Then the sequence
(1011, 1011.1, 1011.101, 1011.10101, ...)
will be an appropriate Zeno sequence of rationals. Their successive
differences are: 0.1, 0.001, 0.00001, ... which obviously satisfies the
criterion for a Zeno sequence, and the differences between each point
and the limit are: .1010100..., .0010100..., .0000100..., which are
clearly no more than twice the corresponding difference.
In general, given a positive irrational number r, define the following
sequence:
r_0 = [r] = greatest integer < r.
r_{n+1} = r_n + 2^m_n
where m_n is the integer for which
r_n + 2^m_n < r < r_n + 2^{m_n+1}
The sequence (m_0, m_1, m_2, ...) is strictly increasing by construction,
therefore we have a Zeno sequence of real numbers.
Define C_n = [A, r_n, B], where (r_n: n >= 0) is the Zeno sequence
constructed for r. Then C_n C_{n+1} = 2^m_n AB (using THEOREM 4), so that
the sequence of points is also a Zeno sequence. Therefore by using AXIOM 8
we can define a point C such that
C C_n <= 2 C_n C_{n+1} = 2^{m_n + 1} AB
By construction, we have
r_n + 2^m_n < r < r_n + 2 2^m_n
so that
r - 2 2^m_n < r_n < r - 2^m_n
Using AXIOM 4 (the Triangle Inequality), we can write:
A C_n - C C_n <= AC <= A C_n + C C_n
(r_n - 2 2^m_n) AB <= AC <= (r_n + 2 2^m_n) AB
(r - 4 2^m_n) AB <= AC <= (r + 2^m_n) AB
Since n is arbitrary, and 2^m_n can be made arbitrarily small (since
m_n is increasing in n), then it follows that AC = r AB.
Similarly, we have
B C_n - C C_n <= BC <= B C_n + C C_n
(|1 - r_n| - 2 2^m_n) AB <= BC <= (|1 - r_n| + 2 2^m_n) AB
(|1 - r| - 4 2^m_n) AB <= BC <= (|1 - r| + 4 2^m_n) AB
so by a similar argument, we establish that BC = |1 - r| AB. Thus C is
uniquely identified as the point [A, r, B].
As you can see above, AXIOM 7 is responsible for much of the underlying
structure given by the theorems just proven, particularly, for showing the
linearity of the set { [A, r, B]: r is a real number } with respect to
the distance measure. That this axiom is a generalization of the
Pythagorean Theorem can be seen by the following
(3) The Pythagorean Theorem
DEFINITIONS: Define an angle to be any sequence of 3 distinct points,
writing "angle ABC" for (A, B, C).
Angle ABC is a right angle if there is a point D such that
AB = BD = AD/2, AC = CD
(i.e. if triangle ABC is half of an isosceles triangle).
THEOREM 11: Pythagorean Theorem
Let A, B and C be distinct points. Then angle ABC is a right
angle if and only if AB^2 + BC^2 = AC^2.
Proof:
Suppose angle ABC is a right angle and point D is chosen so that
AB = BD = AD/2 and AC = CD. Then since AB + BD = AD, by AXIOM 7 we can
write:
AB CD^2 - AD BC^2 + BD AC^2 = AB AD BD
or
AB AC^2 - AD BC^2 + AB AC^2 = AB^2 AD
fron which we get:
AD AC^2 = (AB + AB) AC^2 = AD (AB^2 + BC^2)
Since AB > 0 by hypothesis, it follows that AD > 0 so that we can cancel
out the AD, thus arriving at AC^2 = AB^2 + BC^2.
Suppose A, B and C are three points with AB^2 + BC^2 = AC^2. Let
D = [A, 2, B], so that AB = BD = AD/2. Then using AXIOM 7 again, we
obtain:
AB CD^2 - AD BC^2 + BD AC^2 = AB AD BD
or
AB CD^2 = 2 AB BC^2 - AB (AB^2 + BC^2) + 2 AB^3
= AB (AB^2 + BC^2)
= AB AC^2
Again, assuming that A and B are distinct points, then AB > 0 and we can
cancel out AB to arrive at CD = AC. Therefore, angle ABC is a right angle.
(4) Representation of Inner Product Spaces
Finally, with this structure, we can show that the metric given by the
axioms provides us with a natural inner product. For this we will use the
preliminary results:
DEFINITION: An inner product space is a vector space V with a bilinear map
<V, V> -> R such that:
(i) <0, 0> = 0
(ii) <a, a> > 0 unless a = 0
(iii) <a, b> = <b, a>
THEOREM 12: An inner product space is characterised as a set V with the
following:
(a) 0 is in V
(b) If a, b are in V, then a + b is in V.
(c) If r is in R, a in V, then ra is in V.
(d) If a, b are in V, then <a, b> is in R
subject to the identities:
(1) <ra, b> = r <a, b>
(2) <a+b, c> = <a, c> + <b, c>
(3) 0a = 0
(4) <a, b> = <b, a>
(5) <a - b, a - b> = 0 if and only if a = b
where a - b is defined as a + (-1)b.
Proof:
To prove the identities (a+b)+c = a+(b+c), a+b = b+a, a+0 = a,
(rs)a = r(sa), (r+s)a = ra+sa, 1a = a, a-a = 0 = -a+a, r(a+b) = ra+rb
we only need to apply property (5), using (1)-(4) to reduce the
inner product to 0. Examples:
<(a+0)-a, (a+0)-a> = <a+0,a+0> - 2<a+0,a> + <a,a>
= <0,0>
= <0a,0a> = 0<a,a> = 0
<r(sa)-(rs)a, r(sa)-(rs)a> = <r(sa),r(sa)> - 2<r(sa),(rs)a> + <(rs)a,(rs)a>
= r^2<sa,sa> - 2rrs<sa,a> + (rs)^2 <a,a>
= (rs)^2 <a,a> - 2(rs)^2 <a,a> + (rs)^2 <a,a>
= 0
The others are left as an exercise.
DEFINITION: Let O be a point. Define V_O as the space consisting of the
following operations:
<A,B> = (OA^2 + OB^2 - AB^2)/2
r A = [0, r, A]
A + B = 2 [A, 1/2, B]
THEOREM 13: For each point O, V_O is an inner product space with 0 = O.
Proof:
The properties 0a = [O, 0, A] = O, <A,B> = <B,A> are obvious. If <A,A>
is 0, then OA^2 + OA^2 = AA^2 = 0, or OA^2 = 0. Therefore A = 0. Conversely,
<O,O> = (OO^2 + OO^2 - OO^2)/2 = 0.
Next, we can prove that A - B = O <==> A = B:
A - B = 0
<==> [O, 2, [A, 1/2, -B]] = O
<==> [A, 1/2, -B] = O
<==> [A, 2, O] = -B = [O, -1, B] = [B, 2, O]
<==> A = B
(i.e., [A, 2, O] = C = [B, 2, O] <==> [O, -1, A] = C = [O, -1, B]
<==> A = [O, -1, C] = B)
Therefore, we have property (5).
Finally, to show (1) and (2), we argue as follows. If C = rA, then
<C, B> = (OC^2 + OB^2 - BC^2)/2. But using COROLLARY 3, we have
C = [O, r, A] => OC^2 = (1-r) OO^2 + r OA^2 - r(1-r) OA^2
= r^2 OA^2
BC^2 = (1-r) OB^2 + r AB^2 - r(1-r) OA^2
Thus
<C, B> = (r^2 OA^2 + OB^2 - (1-r) OB^2 - r AB^2 + r(1-r) OA^2)/2
= r (OA^2 + OB^2 - AB^2) = r <A, B>
Thus
<rA, B> = r <A, B>.
Finally, if D = A + B, and E = [A, 1/2, B], then
<D, C> = 2 <E, C> = (OE^2 + OC^2 - CE^2)
Again, using COROLLARY 3, we have:
E = [A, 1/2, B] => OE^2 = 1/2 (OA^2 + OB^2) - 1/4 AB^2
CE^2 = 1/2 (AC^2 + BC^2) - 1/4 AB^2
Therefore
<D, C> = 1/2 (OA^2 + OB^2 - AC^2 - BC^2) + OC^2
= (OA^2 + OC^2 - AC^2)/2 + (OB^2 + OC^2 - BC^2)/2
= <A, C> + <B, C>
Therefore
<A + B, C> = <A, C> + <B, C>
which proves that V_O is an inner product space. Noting that <O. O> = 0,
it also follows that O = 0 in V_O.
Thus, AXIOMS 1-8 characterise an inner product space: i.e., a Euclidean
geometry.
THEOREM 14: In any inner product space V_O, the following hold:
(a) <[A, r, B], D> = (1 - r) <A, D> + r <B, D>
= <(1 - r) A + r B, D>
(b) [A, r, B] = (1 - r) A + r B
(c) AB^2 = <A - B, A - B>
Proof:
Let C = [A, r, B]. Then by COROLLARY 3, we have:
CD^2 = (1 - r) AD^2 + r BD^2 - r(1 - r) AB^2
CO^2 = (1 - r) AO^2 + r BO^2 - r(1 - r) AB^2
DO^2 = (1 - r) DO^2 + r DO^2
Therefore
(CO^2 + DO^2 - CD^2) = (1 - r) (AO^2 + DO^2 - AD^2)
+ r (BO^2 + DO^2 - BD^2)
or
<C, D> = (1 - r) <A, D> + r <B, D>
Part (b) follows from this. Let C' = (1 - r) A + r B. Then
<C - C', C> = <C - C', C'>, by (a)
Therefore
<C - C', C - C'> = 0
C = C'
Finally, to establish part (c) we note that
<A - B, A - B> = <A, A> - 2 <A, B> + <B, B>
= OA^2 - (OA^2 + OB^2 - AB^2) + OB^2
= AB^2
Carefully noting the derivations presented above, we also have the
result
COROLLARY 14: AXIOMS 1-4, 7 and THEOREM 10 characterise an inner product
space.
(5) Dimension
Now we can use AXIOMS 9 and 10 to pin down the dimension of the
Euclidean space to 3. But before this, it will help to illustrate matters
if we work in the more general setting, without these two axioms, to
show the relation between dimension and equilateral sets of points.
DEFINITION: An n-simplex is a set { A_0, A_1, ..., A_n } of n+1 distinct
points. It is called EQUILATERAL if all of its points are of
equal distance from one another.
THEOREM 15: Let V denote an inner product space with an orthonormal basis
{ e_1, e_2, ..., e_n }, for n > 0. Then
{ e, e_1, e_2, ..., e_n } is an equilateral simplex
where e = r_n (e_1 + e_2 + ... + e_n)
r_n = (1 + (n+1)^{1/2})/n
Proof:
First, it is easy to see that (e_i e_j)^2 = 2, for i and j distinct.
Second, let e be as given above, then:
(e e_i)^2 = <e - e_i, e - e_i>
= <e, e> - 2 <e, e_i> + <e_i, e_i>
= n r_n^2 - 2 r_n + 1
But the value indicated for r_n is a solution to the equation
n x^2 - 2x + 1 = 1.
THEOREM 16: Let { A_0, A_1, ..., A_n } be an equilateral n-simplex. Then
there exists a point O such that { A_0, ..., A_{n-1} } is an
orthonormal basis in V_O.
Proof:
Let A_0, A_1, ..., A_n be an equilateral n-simplex, with distances
A_i A_j = r, for i and j distinct. Define the following series of
points:
C_0 = A_0
C_{j+1} = [C_j, 1/(j+2), A_{j+1}], 0 <= j < N - 1
Then we can show:
j 2
(A_i C_j)^2 = ------ r if i <= j
2(j+1)
j+2 2
= ------ r if i > j
2(j+1)
by the following inductive argument. The case where j = 0 is trivial
since (A_i C_0)^2 = (A_i A_0)^2 = r^2 if i > 0, or 0 if i = 0. So
assume the result holds for a given j. Then
2 j+1 2 1 2 j + 1 2
(A C ) = --- (A C ) + --- (A A ) - ------- (C A )
i j+1 j+2 i j j+2 i j+1 (j+2)^2 j j+1
If i <= j the right hand side becomes:
j 2 1 2 1 2 j+1 2
------ r + --- r - ------ r = ------ r
2(j+2) j+2 2(j+2) 2(j+2)
If i = j + 1, then it becomes:
1 2 1 2 1 2 j+1 2
- r + --- 0 - ------ r = ------ r
2 j+2 2(j+2) 2(j+2)
Finally, if i > j + 1, then it becomes:
1 2 1 2 1 2 j+3 2
- r + --- r - ------ r = ------ r
2 j+2 2(j+2) 2(j+2)
which establishes the inductive step.
Finally, let O = [C_N, -R, A_N], where R is to be determined below.
Then for i < N we have:
(A_i O)^2 = (1+R) (A_i C_N)^2 - R (A_i A_N)^2 + R(1+R) (C_N A_N)^2
2 N 2 2
= (1+R) ------ r - R r
2(N+1)
and for i < j < N:
<A_i, A_j> = ( (O A_i)^2 + (O A_j)^2 - (A_i A_j)^2)/2
2 N 2 2 1 2
= (1+R) ------ r - R r - - r
2(N+1) 2
which is 0, provided we choose R so that N(1+R)^2 - (N+1) (2R+1) = 0, or
N R^2 - 2R - 1 = 0
which, not too coincidentally, has R = r_N as a solution.
COROLLARY 17: AXIOMS 1-10 uniquely characterise a 3-dimensional Euclidean
space.
Closely noting the derivations that have been presented, we also have
the resolt:
(6) The Equivalence of Zeno Sequences and Cauchy Sequences
As an additional note, we can also show that Zeno sequences provide a
simpler equivalent to Cauchy sequences for metric spaces. For the sake
of completeness the latter is defined and some of its basic properties
are listed below. The following developments are assumed to be carried out
with AXIOMS 1-4 being given.
In the following, which is self-contained, theorems will be numbered anew.
DEFINITIONS: A function a: n |-> a_n defined on { 0, 1, 2, ... } is called
a SEQUENCE, and will be written as (a_n: n >= 0), or just (a_n).
If f: n |-> fn is an increasing function of n, then (a_fn) is
a SUBSEQUENCE of (a_n).
A CAUCHY sequence is one which, for all e > 0 there is an M >= 0
such that am an < e whenever m, n >= M.
We'll write a_n -> a (and call a the LIMIT of (a_n)), if
for all e > 0 and N >= 0, there is an n >= N such that a_n a < e.
The definition of convergence is weaker than usual, but is equivalent for
Cauchy sequences as is seen by the following:
THEOREM 1: If (a_n) is a Cauchy sequence, then a_n -> a if and only if
for all e > 0, there is an N >= 0, such that
a_n a < e, whenever n >= N
Proof:
Suppose a_n -> a. Let e > 0 be given, and M >= 0 be such that
m, n >= M -> a_m a_n < e/2
We can find an N >= M such that a_N a < e/2. Therefore for all n >= N:
a_n a <= a_n a_N + a_N a < e/2 + e/2 = e
THEOREM 2: If (a_n) is a Cauchy sequence, with a_n -> a, a_n -> a', then
a = a'.
Proof:
Let e > 0 be given. Then we can find an M such that the following hold:
am an < e/3 for m, n >= M
aN a < e/3 for some N >= M
aP a' < e/3 for some P >= M
Therefore
a a' <= a aN + aN aP + aP a < e/3 + e/3 + e/3 = e
Since e > 0 was chosen arbitrarily, it follows that a a' = 0, so that
a = a'.
THEOREM 3: If (a_n: n >= 0) is a Cauchy sequence and (b_n: n >= 0) any
of its subsequences, then (b_n) is also a Cauchy sequence
Further, (a_n) -> a if and only if (b_n) -> a.
Proof:
Let b_n be given by a_fn, with fn an increasing function of n. Then
for any e > 0, let N be such that for m, n > N: a_m a_n < e. Then
fm >= m >= N, fn >= n >= N,
therefore b_m b_n = a_fm a_fn < e
Therefore, (b_n) is a Cauchy sequence as well.
Further, if (a_n) -> a, then again let e > 0, N >= 0 be given such
that
m, n >= N -> a_m a_n < e/2
And let P >= N be given such that a_P a < e/2. Then for any p >= N, we
have
fp >= fN >= N.
Therefore
b_p a = a_fp a <= a_fp a_P + a_P a < e/2 + e/2 = e
which proves that (b_n) -> a.
Finally, suppose that (b_n) -> a. Then for any e > 0 we can find an
M >= 0 such that
m, n >= M -> a_m a_n < e.
and an N >= M such that b_N a < e/2. Then for all n >= M it follows that
a_n a <= a_n a_fN + a_fN a
< e/2 + e/2 = e
Therefore a_n -> a.
In the following we define a Zeno Sequence and Zeno-limits.
DEFINITIONS: A Zeno sequence (a_n) is one for which
a_{n+1} a_{n+2} <= 1/2 a_n a_{n+1}, for all n >= 0
We write a_n ->> a (it's Zeno Limit) if
a_n a <= 2 a_n a_{n+1} for all n >= 0
THEOREM 4: If (a_n) is a Zeno sequence, then
(a) a_n a_{n+1} <= 2^{-(n-m)} a_m a_{m+1} for n >= m >= 0
(b) a_n a_{n+p} <= 2 a_n a_{n+1} for n, p >= 0
Proof:
Part (a) follows by an easy induction:
If n = m: a_m a_{m+1} = 2^{-(m-m)} a_m a_{m+1}
If n >= m such that condition (a) holds for n, then
a_{n+1} a_{n+2} <= 1/2 a_n a_{n+1}
<= 1/2 2^{-(n-m)} a_m a_{m+1}
= 2^{-(n+1-m)} a_m a_{m+1}
For part (b), we use the Triangle Inequality:
a_n a_{n+p} <= a_n a_{n+1} + ... + a_{n+p-1} a_{n+p}
and then use the result of part (a) to get:
a_n a_{n+p} <= sum (a_{n+i} a_{n+i+1}: n <= i < n+p }
<= sum (2^{-i) a_n a_{n+1}: n <= i < n+p }
= 2(1 - 2^{-p}) a_n a_{n+1}
< 2 a_n a_{n+1}
THEOREM 5: Every Zeno sequence is a Cauchy sequence, and for any Zeno
sequence (a_n): a_n -> a if and only if a_n ->> a.
Proof:
Suppose (a_n) is a Zeno sequence, and let r = a_0 a_1. Then for any e > 0,
we can find an N such that r < e 2^N/2. For all m >= n >= N, we have
a_m a_n <= 2 a_n a_{n+1}
<= 2 2^{-n} a_0 a_1
<= 2 2^{-N} r < e
Therefore (a_n) is Cauchy.
Second, suppose a_n ->> a. Let e and N be given as above. Then for all
n >= N, we have
a_n a <= 2 a_n a_{n+1}
<= 2 2^{-n} a_0 a_1
<= 2 2^{-N} r < e
Therefore a_n -> a.
Finally, suppose a_n -> a. Let e > 0 and n >= 0 be given and let N >= n
be such that a_N a < e. Then
a_n a <= a_n a_N + a_N a <= 2 a_n a_{n+1} + e
Since e can be made arbitrarily small, it follows that a_n a <= 2 a_n a_{n+1}.
Therefore a_n ->> a.
We can show that every Cauchy sequence has a Zeno subsequence by
considering the concept of sequences of bounded variation.
DEFINITION: A sequence (a_n) has BOUNDED VARIATION if the set {a_m a_n} of
distances is bounded above.
THEOREM 6: If (a_n) and (b_n) have bounded variation, then the set
{a_m b_n: m, n >= 0 } is bounded above
Proof:
Let A and B be respective bounds for (a_n) and (b_n). Then
a_m b_n <= a_m a_0 + a_0 b_0 + b_0 b_n
<= A + a_0 b_0 + B
Therefore, the set { a_m b_n } is bounded above by A + a_0 b_0 + B.
DEFINITION: If (a_n) and (b_n) are sequences with bounded variation, define
(a_n) (b_n) = inf (sup { a_n b_n: n >= m }: m >= 0)
If (a_n) = (a, a, a, ...) then let a (b_n) = (a_n) (b_n), and
if (b_n) = (b, b, b, ...) then let (a_n) b = (a_n) (b_n).
THEOREM 7: All Zeno sequences have bounded variation
Proof:
If (a_n) is a Zeno sequence, with 0 <= m <= n, then
a_m a_n <= 2 a_m a_{m+1} <= 2 2^{-m} a_0 a_1
Therefore { a_m a_n } is bounded above by 2 a_0 a_1.
THEOREM 8: All Cauchy sequences have bounded variation
Proof:
If (a_n) is a Cauchy sequence, suppose M >= 0 is chosen so that
a_m a_n < 1 whenever m, n >= M
Define
R = max { 1, a_0 a_M, a_1 a_M, ..., a_{M-1} a_M }
Then we can write the following:
If M <= m, n: a_m a_n < 1 <= R <= 2R
If m < M <= n: a_m a_n <= a_m a_M + a_M a_n <= R + 1 <= 2R
If m, n <= M: a_m a_n <= a_m a_M + a_M a_n <= R + R = 2R
Therefore, 2R is an upper bound to {a_m a_n}.
THEOREM 9: Every Cauchy sequence has a Zeno subsequence.
Proof:
Let (a_n) be a Cauchy sequence. Define fn by the following:
f0 = 0
f(k+1) = the least N >= 0 such that
a_m a_n <= 1/3 a_fk (a_n), for m, n >= N
Define b_k = a_fk, and D_k = b_k (a_n). Then
b_{k+1} b_{k+2} = a_{f(k+1)} a_{f(k+2)}
<= 1/3 a_fk (a_n) = 1/3 D_k
but
sup { b_k a_p: p >= f(k+1) } >= D_k
and for all p >= f(k+1), we have
b_k a_p <= b_k b_{k+1} + b_{k+1} a_p <= b_k b_{k+1} + 1/3 D_k
Therefore
b_k b_{k+1} + 1/3 D_k
>= sup { b_k a_p: p >= f(k+1) }
>= D_k
or
b_k b_{k+1} >= 2/3 D_k >= 2 b_{k+1} b_{k+2}
Therefore, (b_k) is a Zeno subsequence of (a_n).
Comparing the corresponding results between Zeno sequences and Cauchy
sequences (e.g., THEOREM 7 and 8) shows just how much simplification can
be achieved by using the former.
As a final result we have the following:
THEOREM 10: Suppose AXIOMS 1-4 hold in a space V (i.e., let V be a Metric
space). Then AXIOM 8 is equivalent to the following condition:
Every Cauchy sequence of points in V has a limit
Proof:
Suppose the condition holds true. Let (a_n) be a Zeno sequence. By
THEOREM 5a, (a_n) is a Cauchy sequence. Our condition grants the existence
of a point, a, such that a_n -> a. By THEOREM 5b, a_n ->> a. Therefore,
AXIOM 8 holds.
Suppose AXIOM 8 is true. Let (a_n) be a Cauchy sequence. By THEOREM 9,
we can select out a Zeno subsequence (b_k). Using AXIOM 8, we can find a
point, a, such that b_k ->> a. By THEOREM 3a (or THEOREM 5a), (b_k) is a
Cauchy sequence, therefore by THEOREM 5b, b_k -> a, and by THEOREM 3b
a_n -> a. This proves that every Cauchy sequence has a limit.
COROLLARY 11: AXIOMS 1-4, 8 characterise a closed metric space.
AXIOMS 1-4, 7-8 and THEOREM 10 of section (2) characterise
a Hilbert Space.
In the following article, I will show how one can carry out the
construction of a closed metric space from a given metric space using Zeno
sequences, and will then relate the axiomitization presented above to
Banach spaces, providing a characterization of the latter.