:: by Jan Popio{\l}ek

::

:: Received July 19, 1991

:: Copyright (c) 1991-2012 Association of Mizar Users

begin

deffunc H

:: deftheorem Def1 defines Cauchy BHSP_3:def 1 :

for X being RealUnitarySpace

for seq being sequence of X holds

( seq is Cauchy iff for r being Real st r > 0 holds

ex k being Element of NAT st

for n, m being Element of NAT st n >= k & m >= k holds

dist ((seq . n),(seq . m)) < r );

for X being RealUnitarySpace

for seq being sequence of X holds

( seq is Cauchy iff for r being Real st r > 0 holds

ex k being Element of NAT st

for n, m being Element of NAT st n >= k & m >= k holds

dist ((seq . n),(seq . m)) < r );

theorem :: BHSP_3:2

for X being RealUnitarySpace

for seq being sequence of X holds

( seq is Cauchy iff for r being Real st r > 0 holds

ex k being Element of NAT st

for n, m being Element of NAT st n >= k & m >= k holds

||.((seq . n) - (seq . m)).|| < r )

for seq being sequence of X holds

( seq is Cauchy iff for r being Real st r > 0 holds

ex k being Element of NAT st

for n, m being Element of NAT st n >= k & m >= k holds

||.((seq . n) - (seq . m)).|| < r )

proof end;

theorem :: BHSP_3:3

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds

seq1 + seq2 is Cauchy

for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds

seq1 + seq2 is Cauchy

proof end;

theorem :: BHSP_3:4

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds

seq1 - seq2 is Cauchy

for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds

seq1 - seq2 is Cauchy

proof end;

theorem Th5: :: BHSP_3:5

for X being RealUnitarySpace

for a being Real

for seq being sequence of X st seq is Cauchy holds

a * seq is Cauchy

for a being Real

for seq being sequence of X st seq is Cauchy holds

a * seq is Cauchy

proof end;

theorem Th7: :: BHSP_3:7

for X being RealUnitarySpace

for x being Point of X

for seq being sequence of X st seq is Cauchy holds

seq + x is Cauchy

for x being Point of X

for seq being sequence of X st seq is Cauchy holds

seq + x is Cauchy

proof end;

theorem :: BHSP_3:8

for X being RealUnitarySpace

for x being Point of X

for seq being sequence of X st seq is Cauchy holds

seq - x is Cauchy

for x being Point of X

for seq being sequence of X st seq is Cauchy holds

seq - x is Cauchy

proof end;

:: deftheorem Def2 defines is_compared_to BHSP_3:def 2 :

for X being RealUnitarySpace

for seq1, seq2 being sequence of X holds

( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq1 . n),(seq2 . n)) < r );

for X being RealUnitarySpace

for seq1, seq2 being sequence of X holds

( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq1 . n),(seq2 . n)) < r );

theorem Th11: :: BHSP_3:11

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is_compared_to seq2 holds

seq2 is_compared_to seq1

for seq1, seq2 being sequence of X st seq1 is_compared_to seq2 holds

seq2 is_compared_to seq1

proof end;

definition

let X be RealUnitarySpace;

let seq1, seq2 be sequence of X;

:: original: is_compared_to

redefine pred seq1 is_compared_to seq2;

reflexivity

for seq1 being sequence of X holds (X,b_{1},b_{1})
by Th10;

symmetry

for seq1, seq2 being sequence of X st (X,b_{1},b_{2}) holds

(X,b_{2},b_{1})
by Th11;

end;
let seq1, seq2 be sequence of X;

:: original: is_compared_to

redefine pred seq1 is_compared_to seq2;

reflexivity

for seq1 being sequence of X holds (X,b

symmetry

for seq1, seq2 being sequence of X st (X,b

(X,b

theorem :: BHSP_3:12

for X being RealUnitarySpace

for seq1, seq2, seq3 being sequence of X st seq1 is_compared_to seq2 & seq2 is_compared_to seq3 holds

seq1 is_compared_to seq3

for seq1, seq2, seq3 being sequence of X st seq1 is_compared_to seq2 & seq2 is_compared_to seq3 holds

seq1 is_compared_to seq3

proof end;

theorem :: BHSP_3:13

for X being RealUnitarySpace

for seq1, seq2 being sequence of X holds

( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

||.((seq1 . n) - (seq2 . n)).|| < r )

for seq1, seq2 being sequence of X holds

( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

||.((seq1 . n) - (seq2 . n)).|| < r )

proof end;

theorem :: BHSP_3:14

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st ex k being Element of NAT st

for n being Element of NAT st n >= k holds

seq1 . n = seq2 . n holds

seq1 is_compared_to seq2

for seq1, seq2 being sequence of X st ex k being Element of NAT st

for n being Element of NAT st n >= k holds

seq1 . n = seq2 . n holds

seq1 is_compared_to seq2

proof end;

theorem :: BHSP_3:15

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is Cauchy & seq1 is_compared_to seq2 holds

seq2 is Cauchy

for seq1, seq2 being sequence of X st seq1 is Cauchy & seq1 is_compared_to seq2 holds

seq2 is Cauchy

proof end;

theorem :: BHSP_3:16

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is convergent & seq1 is_compared_to seq2 holds

seq2 is convergent

for seq1, seq2 being sequence of X st seq1 is convergent & seq1 is_compared_to seq2 holds

seq2 is convergent

proof end;

theorem :: BHSP_3:17

for X being RealUnitarySpace

for g being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g & seq1 is_compared_to seq2 holds

( seq2 is convergent & lim seq2 = g )

for g being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g & seq1 is_compared_to seq2 holds

( seq2 is convergent & lim seq2 = g )

proof end;

:: deftheorem Def3 defines bounded BHSP_3:def 3 :

for X being RealUnitarySpace

for seq being sequence of X holds

( seq is bounded iff ex M being Real st

( M > 0 & ( for n being Element of NAT holds ||.(seq . n).|| <= M ) ) );

for X being RealUnitarySpace

for seq being sequence of X holds

( seq is bounded iff ex M being Real st

( M > 0 & ( for n being Element of NAT holds ||.(seq . n).|| <= M ) ) );

theorem Th18: :: BHSP_3:18

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is bounded & seq2 is bounded holds

seq1 + seq2 is bounded

for seq1, seq2 being sequence of X st seq1 is bounded & seq2 is bounded holds

seq1 + seq2 is bounded

proof end;

theorem :: BHSP_3:20

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is bounded & seq2 is bounded holds

seq1 - seq2 is bounded

for seq1, seq2 being sequence of X st seq1 is bounded & seq2 is bounded holds

seq1 - seq2 is bounded

proof end;

theorem :: BHSP_3:21

for X being RealUnitarySpace

for a being Real

for seq being sequence of X st seq is bounded holds

a * seq is bounded

for a being Real

for seq being sequence of X st seq is bounded holds

a * seq is bounded

proof end;

theorem Th23: :: BHSP_3:23

for X being RealUnitarySpace

for seq being sequence of X

for m being Element of NAT ex M being Real st

( M > 0 & ( for n being Element of NAT st n <= m holds

||.(seq . n).|| < M ) )

for seq being sequence of X

for m being Element of NAT ex M being Real st

( M > 0 & ( for n being Element of NAT st n <= m holds

||.(seq . n).|| < M ) )

proof end;

theorem :: BHSP_3:25

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is bounded & seq1 is_compared_to seq2 holds

seq2 is bounded

for seq1, seq2 being sequence of X st seq1 is bounded & seq1 is_compared_to seq2 holds

seq2 is bounded

proof end;

theorem Th26: :: BHSP_3:26

for X being RealUnitarySpace

for seq, seq1 being sequence of X st seq is bounded & seq1 is subsequence of seq holds

seq1 is bounded

for seq, seq1 being sequence of X st seq is bounded & seq1 is subsequence of seq holds

seq1 is bounded

proof end;

theorem Th27: :: BHSP_3:27

for X being RealUnitarySpace

for seq, seq1 being sequence of X st seq is convergent & seq1 is subsequence of seq holds

seq1 is convergent

for seq, seq1 being sequence of X st seq is convergent & seq1 is subsequence of seq holds

seq1 is convergent

proof end;

theorem Th28: :: BHSP_3:28

for X being RealUnitarySpace

for seq1, seq being sequence of X st seq1 is subsequence of seq & seq is convergent holds

lim seq1 = lim seq

for seq1, seq being sequence of X st seq1 is subsequence of seq & seq is convergent holds

lim seq1 = lim seq

proof end;

theorem Th29: :: BHSP_3:29

for X being RealUnitarySpace

for seq, seq1 being sequence of X st seq is Cauchy & seq1 is subsequence of seq holds

seq1 is Cauchy

for seq, seq1 being sequence of X st seq is Cauchy & seq1 is subsequence of seq holds

seq1 is Cauchy

proof end;

theorem :: BHSP_3:30

for X being RealUnitarySpace

for seq being sequence of X

for k, m being Element of NAT holds (seq ^\ k) ^\ m = (seq ^\ m) ^\ k

for seq being sequence of X

for k, m being Element of NAT holds (seq ^\ k) ^\ m = (seq ^\ m) ^\ k

proof end;

theorem :: BHSP_3:31

for X being RealUnitarySpace

for seq being sequence of X

for k, m being Element of NAT holds (seq ^\ k) ^\ m = seq ^\ (k + m)

for seq being sequence of X

for k, m being Element of NAT holds (seq ^\ k) ^\ m = seq ^\ (k + m)

proof end;

theorem Th32: :: BHSP_3:32

for X being RealUnitarySpace

for seq1, seq2 being sequence of X

for k being Element of NAT holds (seq1 + seq2) ^\ k = (seq1 ^\ k) + (seq2 ^\ k)

for seq1, seq2 being sequence of X

for k being Element of NAT holds (seq1 + seq2) ^\ k = (seq1 ^\ k) + (seq2 ^\ k)

proof end;

theorem Th33: :: BHSP_3:33

for X being RealUnitarySpace

for seq being sequence of X

for k being Element of NAT holds (- seq) ^\ k = - (seq ^\ k)

for seq being sequence of X

for k being Element of NAT holds (- seq) ^\ k = - (seq ^\ k)

proof end;

theorem :: BHSP_3:34

for X being RealUnitarySpace

for seq1, seq2 being sequence of X

for k being Element of NAT holds (seq1 - seq2) ^\ k = (seq1 ^\ k) - (seq2 ^\ k)

for seq1, seq2 being sequence of X

for k being Element of NAT holds (seq1 - seq2) ^\ k = (seq1 ^\ k) - (seq2 ^\ k)

proof end;

theorem :: BHSP_3:35

for X being RealUnitarySpace

for a being Real

for seq being sequence of X

for k being Element of NAT holds (a * seq) ^\ k = a * (seq ^\ k)

for a being Real

for seq being sequence of X

for k being Element of NAT holds (a * seq) ^\ k = a * (seq ^\ k)

proof end;

theorem :: BHSP_3:36

for X being RealUnitarySpace

for seq being sequence of X

for k being Element of NAT st seq is convergent holds

( seq ^\ k is convergent & lim (seq ^\ k) = lim seq ) by Th27, Th28;

for seq being sequence of X

for k being Element of NAT st seq is convergent holds

( seq ^\ k is convergent & lim (seq ^\ k) = lim seq ) by Th27, Th28;

theorem :: BHSP_3:37

for X being RealUnitarySpace

for seq, seq1 being sequence of X st seq is convergent & ex k being Element of NAT st seq = seq1 ^\ k holds

seq1 is convergent

for seq, seq1 being sequence of X st seq is convergent & ex k being Element of NAT st seq = seq1 ^\ k holds

seq1 is convergent

proof end;

theorem :: BHSP_3:38

for X being RealUnitarySpace

for seq, seq1 being sequence of X st seq is Cauchy & ex k being Element of NAT st seq = seq1 ^\ k holds

seq1 is Cauchy

for seq, seq1 being sequence of X st seq is Cauchy & ex k being Element of NAT st seq = seq1 ^\ k holds

seq1 is Cauchy

proof end;

theorem :: BHSP_3:39

theorem :: BHSP_3:40

for X being RealUnitarySpace

for seq1, seq2 being sequence of X

for k being Element of NAT st seq1 is_compared_to seq2 holds

seq1 ^\ k is_compared_to seq2 ^\ k

for seq1, seq2 being sequence of X

for k being Element of NAT st seq1 is_compared_to seq2 holds

seq1 ^\ k is_compared_to seq2 ^\ k

proof end;

theorem :: BHSP_3:41

theorem :: BHSP_3:42

definition

let X be RealUnitarySpace;

end;
attr X is complete means :Def4: :: BHSP_3:def 4

for seq being sequence of X st seq is Cauchy holds

seq is convergent ;

for seq being sequence of X st seq is Cauchy holds

seq is convergent ;

:: deftheorem Def4 defines complete BHSP_3:def 4 :

for X being RealUnitarySpace holds

( X is complete iff for seq being sequence of X st seq is Cauchy holds

seq is convergent );

for X being RealUnitarySpace holds

( X is complete iff for seq being sequence of X st seq is Cauchy holds

seq is convergent );

theorem :: BHSP_3:43

for X being RealUnitarySpace

for seq being sequence of X st X is complete & seq is Cauchy holds

seq is bounded

for seq being sequence of X st X is complete & seq is Cauchy holds

seq is bounded

proof end;