defpred S1[ R_eal] means ( ex g being Real st
( $1 = g & ( for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((seq . m) - $1).| < p ) & seq is convergent_to_finite_number ) or ( $1 = +infty & seq is convergent_to_+infty ) or ( $1 = -infty & seq is convergent_to_-infty ) );
given g1, g2 being R_eal such that A4:
S1[g1]
and
A5:
S1[g2]
and
A6:
g1 <> g2
; contradiction
per cases
( seq is convergent_to_finite_number or seq is convergent_to_+infty or seq is convergent_to_-infty )
by A1;
suppose A7:
seq is
convergent_to_finite_number
;
contradictionthen consider g being
Real such that A8:
g1 = g
and A9:
for
p being
Real st
0 < p holds
ex
n being
Nat st
for
m being
Nat st
n <= m holds
|.((seq . m) - g1).| < p
and
seq is
convergent_to_finite_number
by A4, Th50, Th51;
consider h being
Real such that A10:
g2 = h
and A11:
for
p being
Real st
0 < p holds
ex
n being
Nat st
for
m being
Nat st
n <= m holds
|.((seq . m) - g2).| < p
and
seq is
convergent_to_finite_number
by A5, A7, Th50, Th51;
reconsider g =
g,
h =
h as
Complex ;
g - h <> 0
by A6, A8, A10;
then A12:
|.(g - h).| > 0
;
then consider n1 being
Nat such that A13:
for
m being
Nat st
n1 <= m holds
|.((seq . m) - g1).| < |.(g - h).| / 2
by A9;
consider n2 being
Nat such that A14:
for
m being
Nat st
n2 <= m holds
|.((seq . m) - g2).| < |.(g - h).| / 2
by A11, A12;
reconsider n1 =
n1,
n2 =
n2 as
Element of
NAT by ORDINAL1:def 12;
set m =
max (
n1,
n2);
A15:
|.((seq . (max (n1,n2))) - g1).| < |.(g - h).| / 2
by A13, XXREAL_0:25;
A16:
|.((seq . (max (n1,n2))) - g2).| < |.(g - h).| / 2
by A14, XXREAL_0:25;
reconsider g =
g,
h =
h as
Complex ;
A17:
(seq . (max (n1,n2))) - g2 < |.(g - h).| / 2
by A16, EXTREAL1:21;
A18:
- (|.(g - h).| / 2) < (seq . (max (n1,n2))) - g2
by A16, EXTREAL1:21;
then reconsider w =
(seq . (max (n1,n2))) - g2 as
Element of
REAL by A17, XXREAL_0:48;
A19:
(seq . (max (n1,n2))) - g2 in REAL
by A18, A17, XXREAL_0:48;
then A20:
seq . (max (n1,n2)) <> +infty
by A10;
A21:
(- (seq . (max (n1,n2)))) + g1 = - ((seq . (max (n1,n2))) - g1)
by XXREAL_3:26;
then A22:
|.((- (seq . (max (n1,n2)))) + g1).| < |.(g - h).| / 2
by A15, EXTREAL1:29;
then A23:
(- (seq . (max (n1,n2)))) + g1 < |.(g - h).| / 2
by EXTREAL1:21;
- (|.(g - h).| / 2) < (- (seq . (max (n1,n2)))) + g1
by A22, EXTREAL1:21;
then A24:
(- (seq . (max (n1,n2)))) + g1 in REAL
by A23, XXREAL_0:48;
A25:
seq . (max (n1,n2)) <> -infty
by A10, A19;
|.(g1 - g2).| =
|.((g1 + 0.) - g2).|
by XXREAL_3:4
.=
|.((g1 + ((seq . (max (n1,n2))) + (- (seq . (max (n1,n2)))))) - g2).|
by XXREAL_3:7
.=
|.((((- (seq . (max (n1,n2)))) + g1) + (seq . (max (n1,n2)))) - g2).|
by A8, A20, A25, XXREAL_3:29
.=
|.(((- (seq . (max (n1,n2)))) + g1) + ((seq . (max (n1,n2))) - g2)).|
by A10, A24, XXREAL_3:30
;
then
|.(g1 - g2).| <= |.((- (seq . (max (n1,n2)))) + g1).| + |.((seq . (max (n1,n2))) - g2).|
by EXTREAL1:24;
then A26:
|.(g1 - g2).| <= |.((seq . (max (n1,n2))) - g1).| + |.((seq . (max (n1,n2))) - g2).|
by A21, EXTREAL1:29;
|.w.| in REAL
by XREAL_0:def 1;
then
|.((seq . (max (n1,n2))) - g2).| in REAL
;
then A27:
|.((seq . (max (n1,n2))) - g1).| + |.((seq . (max (n1,n2))) - g2).| < (|.(g - h).| / 2) + |.((seq . (max (n1,n2))) - g2).|
by A15, XXREAL_3:43;
|.(g - h).| / 2
in REAL
by XREAL_0:def 1;
then
|.(g - h).| / 2
in REAL
;
then
(|.(g - h).| / 2) + |.((seq . (max (n1,n2))) - g2).| < (|.(g - h).| / 2) + (|.(g - h).| / 2)
by A16, XXREAL_3:43;
then A28:
|.((seq . (max (n1,n2))) - g1).| + |.((seq . (max (n1,n2))) - g2).| < (|.(g - h).| / 2) + (|.(g - h).| / 2)
by A27, XXREAL_0:2;
g - h = g1 - g2
by A8, A10, SUPINF_2:3;
then
|.(g - h).| = |.(g1 - g2).|
by EXTREAL1:12;
then
|.(g - h).| < (|.(g - h).| / 2) + (|.(g - h).| / 2)
by A28, A26;
hence
contradiction
;
verum end;