:: Convergent Sequences and the Limit of Sequences
:: by Jaros{\l}aw Kotowicz
::
:: Received July 4, 1989
:: Copyright (c) 1990 Association of Mizar Users

begin

theorem :: SEQ_2:1

canceled;

theorem :: SEQ_2:2

canceled;

theorem :: SEQ_2:3

canceled;

theorem :: SEQ_2:4

canceled;

theorem :: SEQ_2:5

canceled;

theorem :: SEQ_2:6

canceled;

theorem :: SEQ_2:7

canceled;

theorem :: SEQ_2:8

canceled;

theorem Th9: :: SEQ_2:9

for g, r being real number holds
( ( - g < r & r < g ) iff abs r < g )

proof end;

theorem :: SEQ_2:10

canceled;

theorem Th11: :: SEQ_2:11

for g, r being real number st g <> 0 & r <> 0 holds
abs ((g ") - (r ")) = (abs (g - r)) / ((abs g) * (abs r))

proof end;

definition

let f be real-valued Function;
attr f is bounded_above means :Def1: :: SEQ_2:def 1
ex r being real number st
for y being set st y in dom f holds
f . y < r;
attr f is bounded_below means :Def2: :: SEQ_2:def 2
ex r being real number st
for y being set st y in dom f holds
r < f . y;

end;

:: deftheorem Def1 defines bounded_above SEQ_2:def 1 :
for f being real-valued Function holds
( f is bounded_above iff ex r being real number st
for y being set st y in dom f holds
f . y < r );

:: deftheorem Def2 defines bounded_below SEQ_2:def 2 :
for f being real-valued Function holds
( f is bounded_below iff ex r being real number st
for y being set st y in dom f holds
r < f . y );

definition

let seq be Real_Sequence;
A1: dom seq = NAT by SEQ_1:3;
redefine attr seq is bounded_above means :Def3: :: SEQ_2:def 3
ex r being real number st
for n being Element of NAT holds seq . n < r;
compatibility
( seq is bounded_above iff ex r being real number st
for n being Element of NAT holds seq . n < r )

proof end;

redefine attr seq is bounded_below means :Def4: :: SEQ_2:def 4
ex r being real number st
for n being Element of NAT holds r < seq . n;
compatibility
( seq is bounded_below iff ex r being real number st
for n being Element of NAT holds r < seq . n )

proof end;

end;

:: deftheorem Def3 defines bounded_above SEQ_2:def 3 :
for seq being Real_Sequence holds
( seq is bounded_above iff ex r being real number st
for n being Element of NAT holds seq . n < r );

:: deftheorem Def4 defines bounded_below SEQ_2:def 4 :
for seq being Real_Sequence holds
( seq is bounded_below iff ex r being real number st
for n being Element of NAT holds r < seq . n );

definition

let f be complex-valued Function;
attr f is bounded means :: SEQ_2:def 5
ex r being real number st
for y being set st y in dom f holds
abs (f . y) < r;

end;

:: deftheorem defines bounded SEQ_2:def 5 :
for f being complex-valued Function holds
( f is bounded iff ex r being real number st
for y being set st y in dom f holds
abs (f . y) < r );

registration

cluster Relation-like Function-like real-valued bounded -> real-valued bounded_above bounded_below set ;
coherence
for b₁ being real-valued Function st b₁ is bounded holds
( b₁ is bounded_above & b₁ is bounded_below )

proof end;

cluster Relation-like Function-like real-valued bounded_above bounded_below -> real-valued bounded set ;
coherence
for b₁ being real-valued Function st b₁ is bounded_above & b₁ is bounded_below holds
b₁ is bounded

proof end;

end;

theorem :: SEQ_2:12

canceled;

theorem :: SEQ_2:13

canceled;

theorem :: SEQ_2:14

canceled;

theorem Th15: :: SEQ_2:15

for seq being Real_Sequence holds
( seq is bounded iff ex r being real number st
( 0 < r & ( for n being Element of NAT holds abs (seq . n) < r ) ) )

proof end;

theorem Th16: :: SEQ_2:16

for seq being Real_Sequence
for n being Element of NAT ex r being real number st
( 0 < r & ( for m being Element of NAT st m <= n holds
abs (seq . m) < r ) )

proof end;

definition

let seq be Real_Sequence;
attr seq is convergent means :Def6: :: SEQ_2:def 6
ex g being real number st
for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((seq . m) - g) < p;

end;

:: deftheorem Def6 defines convergent SEQ_2:def 6 :
for seq being Real_Sequence holds
( seq is convergent iff ex g being real number st
for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((seq . m) - g) < p );

definition

let seq be Real_Sequence;
assume A1: seq is convergent ;
func lim seq -> real number means :Def7: :: SEQ_2:def 7
for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((seq . m) - it) < p;
existence
ex b₁ being real number st
for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((seq . m) - b₁) < p by A1, Def6;
uniqueness
for b₁, b₂ being real number st ( for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((seq . m) - b₁) < p ) & ( for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((seq . m) - b₂) < p ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines lim SEQ_2:def 7 :
for seq being Real_Sequence st seq is convergent holds
for b₂ being real number holds
( b₂ = lim seq iff for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((seq . m) - b₂) < p );

definition

let f be real-valued Function;
redefine attr f is bounded means :: SEQ_2:def 8
( f is bounded_above & f is bounded_below );
compatibility
( f is bounded iff ( f is bounded_above & f is bounded_below ) ) ;

end;

:: deftheorem defines bounded SEQ_2:def 8 :
for f being real-valued Function holds
( f is bounded iff ( f is bounded_above & f is bounded_below ) );

definition

let seq be Real_Sequence;
:: original: lim
redefine func lim seq -> Real;
coherence
lim seq is Real by XREAL_0:def 1;

end;

theorem :: SEQ_2:17

canceled;

theorem :: SEQ_2:18

canceled;

registration

cluster Function-like V23() V27( NAT , REAL ) -> convergent Element of K11(K12(NAT,REAL));
coherence
for b₁ being Real_Sequence st b₁ is constant holds
b₁ is convergent

proof end;

end;

registration

cluster V4() Relation-like NAT -defined REAL -valued Function-like V23() V26( NAT ) V27( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K11(K12(NAT,REAL));
existence
ex b₁ being Real_Sequence st b₁ is constant

proof end;

end;

theorem Th19: :: SEQ_2:19

for seq, seq9 being Real_Sequence st seq is convergent & seq9 is convergent holds
seq + seq9 is convergent

proof end;

registration

let seq1, seq2 be convergent Real_Sequence;
cluster seq1 + seq2 -> convergent Real_Sequence;
coherence
for b₁ being Real_Sequence st b₁ = seq1 + seq2 holds
b₁ is convergent by Th19;

end;

theorem Th20: :: SEQ_2:20

for seq, seq9 being Real_Sequence st seq is convergent & seq9 is convergent holds
lim (seq + seq9) = (lim seq) + (lim seq9)

proof end;

theorem Th21: :: SEQ_2:21

for r being real number
for seq being Real_Sequence st seq is convergent holds
r (#) seq is convergent

proof end;

registration

let r be real number ;
let seq be convergent Real_Sequence;
cluster r (#) seq -> convergent Real_Sequence;
coherence
for b₁ being Real_Sequence st b₁ = r (#) seq holds
b₁ is convergent by Th21;

end;

theorem Th22: :: SEQ_2:22

for r being real number
for seq being Real_Sequence st seq is convergent holds
lim (r (#) seq) = r * (lim seq)

proof end;

theorem :: SEQ_2:23

for seq being Real_Sequence st seq is convergent holds
- seq is convergent ;

registration

let seq be convergent Real_Sequence;
cluster - seq -> convergent Real_Sequence;
coherence
for b₁ being Real_Sequence st b₁ = - seq holds
b₁ is convergent ;

end;

theorem Th24: :: SEQ_2:24

for seq being Real_Sequence st seq is convergent holds
lim (- seq) = - (lim seq)

proof end;

theorem Th25: :: SEQ_2:25

for seq, seq9 being Real_Sequence st seq is convergent & seq9 is convergent holds
seq - seq9 is convergent

proof end;

registration

let seq1, seq2 be convergent Real_Sequence;
cluster seq1 - seq2 -> convergent Real_Sequence;
coherence
for b₁ being Real_Sequence st b₁ = seq1 - seq2 holds
b₁ is convergent by Th25;

end;

theorem Th26: :: SEQ_2:26

for seq, seq9 being Real_Sequence st seq is convergent & seq9 is convergent holds
lim (seq - seq9) = (lim seq) - (lim seq9)

proof end;

theorem Th27: :: SEQ_2:27

for seq being Real_Sequence st seq is convergent holds
seq is bounded

proof end;

registration

cluster Function-like V27( NAT , REAL ) convergent -> bounded Element of K11(K12(NAT,REAL));
coherence
for b₁ being Real_Sequence st b₁ is convergent holds
b₁ is bounded by Th27;

end;

theorem Th28: :: SEQ_2:28

for seq, seq9 being Real_Sequence st seq is convergent & seq9 is convergent holds
seq (#) seq9 is convergent

proof end;

registration

let seq1, seq2 be convergent Real_Sequence;
cluster seq1 (#) seq2 -> convergent Real_Sequence;
coherence
for b₁ being Real_Sequence st b₁ = seq1 (#) seq2 holds
b₁ is convergent by Th28;

end;

theorem Th29: :: SEQ_2:29

for seq, seq9 being Real_Sequence st seq is convergent & seq9 is convergent holds
lim (seq (#) seq9) = (lim seq) * (lim seq9)

proof end;

theorem Th30: :: SEQ_2:30

for seq being Real_Sequence st seq is convergent & lim seq <> 0 holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
(abs (lim seq)) / 2 < abs (seq . m)

proof end;

theorem Th31: :: SEQ_2:31

for seq being Real_Sequence st seq is convergent & ( for n being Element of NAT holds 0 <= seq . n ) holds
0 <= lim seq

proof end;

theorem :: SEQ_2:32

for seq, seq9 being Real_Sequence st seq is convergent & seq9 is convergent & ( for n being Element of NAT holds seq . n <= seq9 . n ) holds
lim seq <= lim seq9

proof end;

theorem Th33: :: SEQ_2:33

for seq, seq9, seq1 being Real_Sequence st seq is convergent & seq9 is convergent & ( for n being Element of NAT holds
( seq . n <= seq1 . n & seq1 . n <= seq9 . n ) ) & lim seq = lim seq9 holds
seq1 is convergent

proof end;

theorem :: SEQ_2:34

for seq, seq9, seq1 being Real_Sequence st seq is convergent & seq9 is convergent & ( for n being Element of NAT holds
( seq . n <= seq1 . n & seq1 . n <= seq9 . n ) ) & lim seq = lim seq9 holds
lim seq1 = lim seq

proof end;

theorem Th35: :: SEQ_2:35

for seq being Real_Sequence st seq is convergent & lim seq <> 0 & seq is non-empty holds
seq " is convergent

proof end;

theorem Th36: :: SEQ_2:36

for seq being Real_Sequence st seq is convergent & lim seq <> 0 & seq is non-empty holds
lim (seq ") = (lim seq) "

proof end;

theorem :: SEQ_2:37

for seq9, seq being Real_Sequence st seq9 is convergent & seq is convergent & lim seq <> 0 & seq is non-empty holds
seq9 /" seq is convergent

proof end;

theorem :: SEQ_2:38

for seq9, seq being Real_Sequence st seq9 is convergent & seq is convergent & lim seq <> 0 & seq is non-empty holds
lim (seq9 /" seq) = (lim seq9) / (lim seq)

proof end;

theorem Th39: :: SEQ_2:39

for seq, seq1 being Real_Sequence st seq is convergent & seq1 is bounded & lim seq = 0 holds
seq (#) seq1 is convergent

proof end;

theorem :: SEQ_2:40

for seq, seq1 being Real_Sequence st seq is convergent & seq1 is bounded & lim seq = 0 holds
lim (seq (#) seq1) = 0

proof end;