let X be non empty set ; :: thesis: for seq being sequence of st seq is CCauchy holds
seq is convergent

let vseq be sequence of ; :: thesis: ( vseq is CCauchy implies vseq is convergent )
defpred S1[ set , set ] means ex xseq being Complex_Sequence st
for n being Nat holds
( xseq . n = (modetrans ((vseq . n),X)) . \$1 & xseq is convergent & \$2 = lim xseq );
assume A1: vseq is CCauchy ; :: thesis: vseq is convergent
A2: for x being Element of X ex y being Element of COMPLEX st S1[x,y]
proof
let x be Element of X; :: thesis: ex y being Element of COMPLEX st S1[x,y]
deffunc H1( Nat) -> Element of COMPLEX = (modetrans ((vseq . \$1),X)) . x;
consider xseq being Complex_Sequence such that
A3: for n being Element of NAT holds xseq . n = H1(n) from A4: for n being Nat holds xseq . n = H1(n)
proof
let n be Nat; :: thesis: xseq . n = H1(n)
n in NAT by ORDINAL1:def 12;
hence xseq . n = H1(n) by A3; :: thesis: verum
end;
reconsider y = lim xseq as Element of COMPLEX by XCMPLX_0:def 2;
take y ; :: thesis: S1[x,y]
A5: for m, k being Nat holds |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).||
proof
let m, k be Nat; :: thesis: |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).||
A6: ( m in NAT & k in NAT ) by ORDINAL1:def 12;
(vseq . m) - (vseq . k) in ComplexBoundedFunctions X ;
then consider h1 being Function of X,COMPLEX such that
A7: h1 = (vseq . m) - (vseq . k) and
A8: h1 | X is bounded ;
vseq . m in ComplexBoundedFunctions X ;
then ex vseqm being Function of X,COMPLEX st
( vseq . m = vseqm & vseqm | X is bounded ) ;
then A9: modetrans ((vseq . m),X) = vseq . m by Th12;
vseq . k in ComplexBoundedFunctions X ;
then ex vseqk being Function of X,COMPLEX st
( vseq . k = vseqk & vseqk | X is bounded ) ;
then A10: modetrans ((vseq . k),X) = vseq . k by Th12;
( xseq . m = (modetrans ((vseq . m),X)) . x & xseq . k = (modetrans ((vseq . k),X)) . x ) by A3, A6;
then (xseq . m) - (xseq . k) = h1 . x by A9, A10, A7, Th26;
hence |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).|| by A7, A8, Th19; :: thesis: verum
end;
now :: thesis: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
|.((xseq . n) - (xseq . k)).| < e
let e be Real; :: thesis: ( e > 0 implies ex k being Nat st
for n being Nat st n >= k holds
|.((xseq . n) - (xseq . k)).| < e )

assume e > 0 ; :: thesis: ex k being Nat st
for n being Nat st n >= k holds
|.((xseq . n) - (xseq . k)).| < e

then consider k being Nat such that
A11: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by ;
reconsider k = k as Nat ;
take k = k; :: thesis: for n being Nat st n >= k holds
|.((xseq . n) - (xseq . k)).| < e

hereby :: thesis: verum
let n be Nat; :: thesis: ( n >= k implies |.((xseq . n) - (xseq . k)).| < e )
assume n >= k ; :: thesis: |.((xseq . n) - (xseq . k)).| < e
then A12: ||.((vseq . n) - (vseq . k)).|| < e by A11;
|.((xseq . n) - (xseq . k)).| <= ||.((vseq . n) - (vseq . k)).|| by A5;
hence |.((xseq . n) - (xseq . k)).| < e by ; :: thesis: verum
end;
end;
then xseq is convergent by COMSEQ_3:46;
hence S1[x,y] by A4; :: thesis: verum
end;
consider tseq being Function of X,COMPLEX such that
A13: for x being Element of X holds S1[x,tseq . x] from
now :: thesis: for e1 being Real st e1 > 0 holds
ex k being Nat st
for m being Nat st m >= k holds
|.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1
let e1 be Real; :: thesis: ( e1 > 0 implies ex k being Nat st
for m being Nat st m >= k holds
|.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1 )

assume A14: e1 > 0 ; :: thesis: ex k being Nat st
for m being Nat st m >= k holds
|.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1

reconsider e = e1 as Real ;
consider k being Nat such that
A15: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by ;
take k = k; :: thesis: for m being Nat st m >= k holds
|.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1

now :: thesis: for m being Nat st m >= k holds
|.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1
let m be Nat; :: thesis: ( m >= k implies |.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1 )
A16: ||.(vseq . m).|| = ||.vseq.|| . m by NORMSP_0:def 4;
assume m >= k ; :: thesis: |.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1
then A17: ||.((vseq . m) - (vseq . k)).|| < e by A15;
( |.(||.(vseq . m).|| - ||.(vseq . k).||).| <= ||.((vseq . m) - (vseq . k)).|| & ||.(vseq . k).|| = ||.vseq.|| . k ) by ;
hence |.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1 by ; :: thesis: verum
end;
hence for m being Nat st m >= k holds
|.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1 ; :: thesis: verum
end;
then A18: ||.vseq.|| is convergent by SEQ_4:41;
now :: thesis: for x being set st x in X /\ (dom tseq) holds
|.(tseq . x).| <= lim ||.vseq.||
let x be set ; :: thesis: ( x in X /\ (dom tseq) implies |.(tseq . x).| <= lim ||.vseq.|| )
assume A19: x in X /\ (dom tseq) ; :: thesis: |.(tseq . x).| <= lim ||.vseq.||
then consider xseq being Complex_Sequence such that
A20: for n being Nat holds xseq . n = (modetrans ((vseq . n),X)) . x and
A21: ( xseq is convergent & tseq . x = lim xseq ) by A13;
A22: for n being Nat holds |.xseq.| . n <= ||.vseq.|| . n
proof
let n be Nat; :: thesis: |.xseq.| . n <= ||.vseq.|| . n
A23: xseq . n = (modetrans ((vseq . n),X)) . x by A20;
vseq . n in ComplexBoundedFunctions X ;
then A24: ex h1 being Function of X,COMPLEX st
( vseq . n = h1 & h1 | X is bounded ) ;
then modetrans ((vseq . n),X) = vseq . n by Th12;
then |.(xseq . n).| <= ||.(vseq . n).|| by ;
then |.xseq.| . n <= ||.(vseq . n).|| by VALUED_1:18;
hence |.xseq.| . n <= ||.vseq.|| . n by NORMSP_0:def 4; :: thesis: verum
end;
( |.xseq.| is convergent & |.(tseq . x).| = lim |.xseq.| ) by ;
hence |.(tseq . x).| <= lim ||.vseq.|| by ; :: thesis: verum
end;
then for x being Element of X st x in X /\ (dom tseq) holds
|.(tseq /. x).| <= lim ||.vseq.|| ;
then tseq | X is bounded by CFUNCT_1:69;
then tseq in ComplexBoundedFunctions X ;
then reconsider tv = tseq as Point of ;
A25: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e
proof
let e be Real; :: thesis: ( e > 0 implies ex k being Nat st
for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e )

assume e > 0 ; :: thesis: ex k being Nat st
for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e

then consider k being Nat such that
A26: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by ;
take k ; :: thesis: for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e

now :: thesis: for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e
let n be Nat; :: thesis: ( n >= k implies for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e )
assume A27: n >= k ; :: thesis: for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e
now :: thesis: for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e
let x be Element of X; :: thesis: |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e
consider xseq being Complex_Sequence such that
A28: for n being Nat holds xseq . n = (modetrans ((vseq . n),X)) . x and
A29: xseq is convergent and
A30: tseq . x = lim xseq by A13;
reconsider xn = xseq . n as Element of COMPLEX by XCMPLX_0:def 2;
reconsider fseq = NAT --> xn as Complex_Sequence ;
set wseq = xseq - fseq;
deffunc H1( Nat) -> object = |.((xseq . \$1) - (xseq . n)).|;
consider rseq being Real_Sequence such that
A31: for m being Nat holds rseq . m = H1(m) from SEQ_1:sch 1();
A32: for m, k being Nat holds |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).||
proof
let m, k be Nat; :: thesis: |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).||
(vseq . m) - (vseq . k) in ComplexBoundedFunctions X ;
then consider h1 being Function of X,COMPLEX such that
A33: h1 = (vseq . m) - (vseq . k) and
A34: h1 | X is bounded ;
vseq . m in ComplexBoundedFunctions X ;
then ex vseqm being Function of X,COMPLEX st
( vseq . m = vseqm & vseqm | X is bounded ) ;
then A35: modetrans ((vseq . m),X) = vseq . m by Th12;
vseq . k in ComplexBoundedFunctions X ;
then ex vseqk being Function of X,COMPLEX st
( vseq . k = vseqk & vseqk | X is bounded ) ;
then A36: modetrans ((vseq . k),X) = vseq . k by Th12;
( xseq . m = (modetrans ((vseq . m),X)) . x & xseq . k = (modetrans ((vseq . k),X)) . x ) by A28;
then (xseq . m) - (xseq . k) = h1 . x by ;
hence |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).|| by ; :: thesis: verum
end;
A37: for m being Nat st m >= k holds
rseq . m <= e
proof
let m be Nat; :: thesis: ( m >= k implies rseq . m <= e )
assume m >= k ; :: thesis: rseq . m <= e
then A38: ||.((vseq . n) - (vseq . m)).|| < e by ;
rseq . m = |.((xseq . m) - (xseq . n)).| by A31;
then rseq . m = |.((xseq . n) - (xseq . m)).| by COMPLEX1:60;
then rseq . m <= ||.((vseq . n) - (vseq . m)).|| by A32;
hence rseq . m <= e by ; :: thesis: verum
end;
A39: now :: thesis: for m being Element of NAT holds (xseq - fseq) . m = (xseq . m) - (xseq . n)
let m be Element of NAT ; :: thesis: (xseq - fseq) . m = (xseq . m) - (xseq . n)
(xseq - fseq) . m = (xseq . m) + ((- fseq) . m) by VALUED_1:1
.= (xseq . m) - (fseq . m) by VALUED_1:8 ;
hence (xseq - fseq) . m = (xseq . m) - (xseq . n) ; :: thesis: verum
end;
now :: thesis: for x being object st x in NAT holds
rseq . x = |.(xseq - fseq).| . x
let x be object ; :: thesis: ( x in NAT implies rseq . x = |.(xseq - fseq).| . x )
assume x in NAT ; :: thesis: rseq . x = |.(xseq - fseq).| . x
then reconsider k = x as Element of NAT ;
rseq . x = |.((xseq . k) - (xseq . n)).| by A31;
then rseq . x = |.((xseq - fseq) . k).| by A39;
hence rseq . x = |.(xseq - fseq).| . x by VALUED_1:18; :: thesis: verum
end;
then A40: rseq = |.(xseq - fseq).| by FUNCT_2:12;
A41: fseq is convergent by CFCONT_1:26;
A42: lim rseq <= e by ;
lim fseq = fseq . 0 by CFCONT_1:28;
then lim fseq = xseq . n ;
then lim (xseq - fseq) = (tseq . x) - (xseq . n) by ;
then lim rseq = |.((tseq . x) - (xseq . n)).| by ;
then |.((xseq . n) - (tseq . x)).| <= e by ;
hence |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e by A28; :: thesis: verum
end;
hence for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e ; :: thesis: verum
end;
hence for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e ; :: thesis: verum
end;
A43: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e
proof
let e be Real; :: thesis: ( e > 0 implies ex k being Nat st
for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e )

assume e > 0 ; :: thesis: ex k being Nat st
for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e

then consider k being Nat such that
A44: for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e by A25;
take k ; :: thesis: for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e

hereby :: thesis: verum
let n be Nat; :: thesis: ( n >= k implies ||.((vseq . n) - tv).|| <= e )
assume A45: n >= k ; :: thesis: ||.((vseq . n) - tv).|| <= e
vseq . n in ComplexBoundedFunctions X ;
then consider f1 being Function of X,COMPLEX such that
A46: f1 = vseq . n and
f1 | X is bounded ;
(vseq . n) - tv in ComplexBoundedFunctions X ;
then consider h1 being Function of X,COMPLEX such that
A47: h1 = (vseq . n) - tv and
A48: h1 | X is bounded ;
A49: now :: thesis: for t being Element of X holds |.(h1 . t).| <= e
let t be Element of X; :: thesis: |.(h1 . t).| <= e
( modetrans ((vseq . n),X) = f1 & h1 . t = (f1 . t) - (tseq . t) ) by ;
hence |.(h1 . t).| <= e by ; :: thesis: verum
end;
A50: now :: thesis: for r being Real st r in PreNorms h1 holds
r <= e
let r be Real; :: thesis: ( r in PreNorms h1 implies r <= e )
assume r in PreNorms h1 ; :: thesis: r <= e
then ex t being Element of X st r = |.(h1 . t).| ;
hence r <= e by A49; :: thesis: verum
end;
( ( for s being Real st s in PreNorms h1 holds
s <= e ) implies upper_bound (PreNorms h1) <= e ) by SEQ_4:45;
hence ||.((vseq . n) - tv).|| <= e by ; :: thesis: verum
end;
end;
for e being Real st e > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e
proof
let e be Real; :: thesis: ( e > 0 implies ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e )

assume A51: e > 0 ; :: thesis: ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e

consider m being Nat such that
A52: for n being Nat st n >= m holds
||.((vseq . n) - tv).|| <= e / 2 by ;
take m ; :: thesis: for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e

A53: e / 2 < e by ;
hereby :: thesis: verum
let n be Nat; :: thesis: ( n >= m implies ||.((vseq . n) - tv).|| < e )
assume n >= m ; :: thesis: ||.((vseq . n) - tv).|| < e
then ||.((vseq . n) - tv).|| <= e / 2 by A52;
hence ||.((vseq . n) - tv).|| < e by ; :: thesis: verum
end;
end;
hence vseq is convergent ; :: thesis: verum