Boschernitsana theorem
- Tutorial
The article provides a simple proof that the mapping of a compact metric space into itself, which does not reduce the distance, is an isometry.
Display
metric space with metric
called isometry if for any
fair equality
. We prove here the following statement:
Recall some simple statements about metric compacts and introduce some conventions and definitions necessary for further discussion.
Through
we denote the number of elements of a finite set
.
For
and
lots of
let's call
- point neighborhood
(or an open ball with a center at
and radius
).
Final set
let's call
-network in
(or simply
network) if for any point
there is a point
such that
. Lots of
let's call
- sparse if
for any
such that
.
For any finite set
denote by
amount of
. Magnitude
call the length of the set
.
1. Let the sequence
,
elements of the set
converge respectively
to the points
. Then
at
.
Proof . Consider obvious inequalities.


Because
,
at
then for
there is such a natural
that for all
will be

Of
follows that
for all
.
2. For each
at
there is an ultimate
-network.
Proof . Open Ball Family
where
runs through
is a coating
. T. to.
compact, choose the final family of balls
also covering
. Clearly a lot
- the ultimate
-network.
3. Space
limited. Namely, there is such a number
, what
for any
.
The proof immediately follows from 2. Indeed, we set
where
,
- elements
-networks
. It's clear that
.
4. If
- the ultimate
- network in
then for any
- sparse set
will be
i.e.
.
Proof . Pool balls covers
. If a
then two different elements from
will be in one of the balls
that contradicts the fact that
-
- sparse set.
5. To each
-resolved set
set the number
- its length. We have already proven that a function that puts any
-resolved set
in line number
is limited. Note that the function that each
-resolved set
matches its length
is also limited.
6. Let
where
taken on all
- sparse sets
. Then fair
Proof . Will consider
-network
from Lemma 1. If
does not belong to the ball
then
not belong
. This means that there is such
, what
and
. Similarly, there is such
, what
and
. Rate
. It's clear that
. And since
and
,
then
. Consequently,
.
So, we proved that
continuously displays
at
. From Lemma 1 it follows that for each
exists
- network in
such that
preserves the distance between the elements of this network. So for any points
can find sequences
,
such that
. But
at
. From the continuity of the display
follows that
,
at
. Consequently,
at
. And since for any
equality is fulfilled
then
.
This proof of the Boschernitsan theorem is based on conversations with my student comrade, now an American mathematician Leonid Luxemburg, during one of his visits to Moscow and is my presentation of the idea he proposed.
Slobodnik Semen Grigorievich ,
content developer for the application “Tutor: Mathematics” (see the article on Habré ), Candidate of Physical and Mathematical Sciences, math teacher of school 179, Moscow
Display
Theorem. If amapping of a compact metric space into itself, such that
for anythen mapping
- isometry.
Recall some simple statements about metric compacts and introduce some conventions and definitions necessary for further discussion.
Through
For
Final set
For any finite set
1. Let the sequence
to the points
Proof . Consider obvious inequalities.
Because
Of
2. For each
Proof . Open Ball Family
3. Space
The proof immediately follows from 2. Indeed, we set
4. If
Proof . Pool balls covers
5. To each
6. Let
Lemma 1. There is-spine set
such that
,
is an
-network in
,
is also
-network in
and for any
will be
.
7. Lemma 2. Mappingcontinuously on
. More precisely: if
for any
then
.
Proof . Will consider
So, we proved that
Comment
This proof of the Boschernitsan theorem is based on conversations with my student comrade, now an American mathematician Leonid Luxemburg, during one of his visits to Moscow and is my presentation of the idea he proposed.

content developer for the application “Tutor: Mathematics” (see the article on Habré ), Candidate of Physical and Mathematical Sciences, math teacher of school 179, Moscow