let X, Y be ComplexNormSpace; :: thesis: ( Y is complete implies for seq being sequence of st seq is Cauchy_sequence_by_Norm holds
seq is convergent )

assume A1: Y is complete ; :: thesis: for seq being sequence of st seq is Cauchy_sequence_by_Norm holds
seq is convergent

let vseq be sequence of ; :: thesis: ( vseq is Cauchy_sequence_by_Norm implies vseq is convergent )
assume A2: vseq is Cauchy_sequence_by_Norm ; :: thesis: vseq is convergent
defpred S1[ set , set ] means ex xseq being sequence of Y st
( ( for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . \$1 ) & xseq is convergent & \$2 = lim xseq );
A3: for x being Element of X ex y being Element of Y st S1[x,y]
proof
let x be Element of X; :: thesis: ex y being Element of Y st S1[x,y]
deffunc H1( Nat) -> Element of the carrier of Y = (modetrans ((vseq . \$1),X,Y)) . x;
consider xseq being sequence of Y such that
A4: for n being Element of NAT holds xseq . n = H1(n) from A5: 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 A4; :: thesis: verum
end;
take lim xseq ; :: thesis: S1[x, lim xseq]
A6: 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)).|| *
A7: k in NAT by ORDINAL1:def 12;
A8: m in NAT by ORDINAL1:def 12;
reconsider h1 = (vseq . m) - (vseq . k) as Lipschitzian LinearOperator of X,Y by Def7;
A9: xseq . k = (modetrans ((vseq . k),X,Y)) . x by A4, A7;
vseq . m is Lipschitzian LinearOperator of X,Y by Def7;
then A10: modetrans ((vseq . m),X,Y) = vseq . m by Th28;
vseq . k is Lipschitzian LinearOperator of X,Y by Def7;
then A11: modetrans ((vseq . k),X,Y) = vseq . k by Th28;
xseq . m = (modetrans ((vseq . m),X,Y)) . x by A4, A8;
then (xseq . m) - (xseq . k) = h1 . x by A10, A11, A9, Th39;
hence ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * by Th31; :: thesis: verum
end;
now :: thesis: for e being Real st e > 0 holds
ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e
let e be Real; :: thesis: ( e > 0 implies ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e )

assume A12: e > 0 ; :: thesis: ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e

now :: thesis: ( ( x = 0. X & ex k being Element of NAT st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e ) or ( x <> 0. X & ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e ) )
per cases ( x = 0. X or x <> 0. X ) ;
case A13: x = 0. X ; :: thesis: ex k being Element of NAT st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e

take k = 0 ; :: thesis: for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e

thus for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e :: thesis: verum
proof
let n, m be Nat; :: thesis: ( n >= k & m >= k implies ||.((xseq . n) - (xseq . m)).|| < e )
assume that
n >= k and
m >= k ; :: thesis: ||.((xseq . n) - (xseq . m)).|| < e
A14: n in NAT by ORDINAL1:def 12;
A15: m in NAT by ORDINAL1:def 12;
A16: xseq . m = (modetrans ((vseq . m),X,Y)) . x by
.= (modetrans ((vseq . m),X,Y)) . (0c * x) by
.= 0c * ((modetrans ((vseq . m),X,Y)) . x) by Def3
.= 0. Y by CLVECT_1:1 ;
xseq . n = (modetrans ((vseq . n),X,Y)) . x by
.= (modetrans ((vseq . n),X,Y)) . (0c * x) by
.= 0c * ((modetrans ((vseq . n),X,Y)) . x) by Def3
.= 0. Y by CLVECT_1:1 ;
then ||.((xseq . n) - (xseq . m)).|| = ||.(0. Y).|| by
.= 0 by NORMSP_0:def 6 ;
hence ||.((xseq . n) - (xseq . m)).|| < e by A12; :: thesis: verum
end;
end;
case x <> 0. X ; :: thesis: ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e

then A17: ||.x.|| <> 0 by NORMSP_0:def 5;
then A18: ||.x.|| > 0 by CLVECT_1:105;
then consider k being Nat such that
A19: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e / by ;
take k = k; :: thesis: for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e

thus for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e :: thesis: verum
proof
let n, m be Nat; :: thesis: ( n >= k & m >= k implies ||.((xseq . n) - (xseq . m)).|| < e )
assume that
A20: n >= k and
A21: m >= k ; :: thesis: ||.((xseq . n) - (xseq . m)).|| < e
||.((vseq . n) - (vseq . m)).|| < e / by ;
then A22: ||.((vseq . n) - (vseq . m)).|| * < (e / ) * by ;
A23: (e / ) * = (e * ()) * by XCMPLX_0:def 9
.= e * (() * )
.= e * 1 by
.= e ;
||.((xseq . n) - (xseq . m)).|| <= ||.((vseq . n) - (vseq . m)).|| * by A6;
hence ||.((xseq . n) - (xseq . m)).|| < e by ; :: thesis: verum
end;
end;
end;
end;
hence ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e ; :: thesis: verum
end;
then xseq is Cauchy_sequence_by_Norm by CSSPACE3:8;
then xseq is convergent by A1;
hence S1[x, lim xseq] by A5; :: thesis: verum
end;
consider f being Function of the carrier of X, the carrier of Y such that
A24: for x being Element of X holds S1[x,f . x] from reconsider tseq = f as Function of X,Y ;
A25: now :: thesis: for x, y being VECTOR of X holds tseq . (x + y) = (tseq . x) + (tseq . y)
let x, y be VECTOR of X; :: thesis: tseq . (x + y) = (tseq . x) + (tseq . y)
consider xseq being sequence of Y such that
A26: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A27: xseq is convergent and
A28: tseq . x = lim xseq by A24;
consider zseq being sequence of Y such that
A29: for n being Nat holds zseq . n = (modetrans ((vseq . n),X,Y)) . (x + y) and
zseq is convergent and
A30: tseq . (x + y) = lim zseq by A24;
consider yseq being sequence of Y such that
A31: for n being Nat holds yseq . n = (modetrans ((vseq . n),X,Y)) . y and
A32: yseq is convergent and
A33: tseq . y = lim yseq by A24;
now :: thesis: for n being Nat holds zseq . n = (xseq . n) + (yseq . n)
let n be Nat; :: thesis: zseq . n = (xseq . n) + (yseq . n)
thus zseq . n = (modetrans ((vseq . n),X,Y)) . (x + y) by A29
.= ((modetrans ((vseq . n),X,Y)) . x) + ((modetrans ((vseq . n),X,Y)) . y) by VECTSP_1:def 20
.= (xseq . n) + ((modetrans ((vseq . n),X,Y)) . y) by A26
.= (xseq . n) + (yseq . n) by A31 ; :: thesis: verum
end;
then zseq = xseq + yseq by NORMSP_1:def 2;
hence tseq . (x + y) = (tseq . x) + (tseq . y) by ; :: thesis: verum
end;
now :: thesis: for x being VECTOR of X
for c being Complex holds tseq . (c * x) = c * (tseq . x)
let x be VECTOR of X; :: thesis: for c being Complex holds tseq . (c * x) = c * (tseq . x)
let c be Complex; :: thesis: tseq . (c * x) = c * (tseq . x)
consider xseq being sequence of Y such that
A34: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A35: xseq is convergent and
A36: tseq . x = lim xseq by A24;
consider zseq being sequence of Y such that
A37: for n being Nat holds zseq . n = (modetrans ((vseq . n),X,Y)) . (c * x) and
zseq is convergent and
A38: tseq . (c * x) = lim zseq by A24;
now :: thesis: for n being Nat holds zseq . n = c * (xseq . n)
let n be Nat; :: thesis: zseq . n = c * (xseq . n)
thus zseq . n = (modetrans ((vseq . n),X,Y)) . (c * x) by A37
.= c * ((modetrans ((vseq . n),X,Y)) . x) by Def3
.= c * (xseq . n) by A34 ; :: thesis: verum
end;
then zseq = c * xseq by CLVECT_1:def 14;
hence tseq . (c * x) = c * (tseq . x) by ; :: thesis: verum
end;
then reconsider tseq = tseq as LinearOperator of X,Y by ;
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 A39: 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
A40: 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 )
assume m >= k ; :: thesis: |.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1
then A41: ||.((vseq . m) - (vseq . k)).|| < e by A40;
A42: ||.(vseq . m).|| = ||.vseq.|| . m by NORMSP_0:def 4;
A43: ||.(vseq . k).|| = ||.vseq.|| . k by NORMSP_0:def 4;
|.(||.(vseq . m).|| - ||.(vseq . k).||).| <= ||.((vseq . m) - (vseq . k)).|| by CLVECT_1:110;
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 A44: ||.vseq.|| is convergent by SEQ_4:41;
A45: tseq is Lipschitzian
proof
take lim ||.vseq.|| ; :: according to CLOPBAN1:def 6 :: thesis: ( 0 <= lim ||.vseq.|| & ( for x being VECTOR of X holds ||.(tseq . x).|| <= (lim ||.vseq.||) * ) )
A46: now :: thesis: for x being VECTOR of X holds ||.(tseq . x).|| <= (lim ||.vseq.||) *
let x be VECTOR of X; :: thesis: ||.(tseq . x).|| <= (lim ||.vseq.||) *
consider xseq being sequence of Y such that
A47: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A48: xseq is convergent and
A49: tseq . x = lim xseq by A24;
A50: ||.(tseq . x).|| = lim ||.xseq.|| by ;
A51: for m being Nat holds ||.(xseq . m).|| <= ||.(vseq . m).|| *
proof
let m be Nat; :: thesis: ||.(xseq . m).|| <= ||.(vseq . m).|| *
A52: xseq . m = (modetrans ((vseq . m),X,Y)) . x by A47;
vseq . m is Lipschitzian LinearOperator of X,Y by Def7;
hence ||.(xseq . m).|| <= ||.(vseq . m).|| * by ; :: thesis: verum
end;
A53: for n being Nat holds ||.xseq.|| . n <= ( (#) ||.vseq.||) . n
proof
let n be Nat; :: thesis: ||.xseq.|| . n <= ( (#) ||.vseq.||) . n
A54: ||.xseq.|| . n = ||.(xseq . n).|| by NORMSP_0:def 4;
A55: ||.(vseq . n).|| = ||.vseq.|| . n by NORMSP_0:def 4;
||.(xseq . n).|| <= ||.(vseq . n).|| * by A51;
hence ||.xseq.|| . n <= ( (#) ||.vseq.||) . n by ; :: thesis: verum
end;
A56: ||.x.|| (#) ||.vseq.|| is convergent by A44;
A57: lim ( (#) ||.vseq.||) = (lim ||.vseq.||) * by ;
||.xseq.|| is convergent by ;
hence ||.(tseq . x).|| <= (lim ||.vseq.||) * by ; :: thesis: verum
end;
now :: thesis: for n being Nat holds ||.vseq.|| . n >= 0
let n be Nat; :: thesis: ||.vseq.|| . n >= 0
||.(vseq . n).|| >= 0 by CLVECT_1:105;
hence ||.vseq.|| . n >= 0 by NORMSP_0:def 4; :: thesis: verum
end;
hence ( 0 <= lim ||.vseq.|| & ( for x being VECTOR of X holds ||.(tseq . x).|| <= (lim ||.vseq.||) * ) ) by ; :: thesis: verum
end;
A58: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . 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 VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * )

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

then consider k being Nat such that
A59: 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 VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e *

now :: thesis: for n being Nat st n >= k holds
for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e *
let n be Nat; :: thesis: ( n >= k implies for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * )
assume A60: n >= k ; :: thesis: for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e *
now :: thesis: for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e *
let x be VECTOR of X; :: thesis: ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e *
consider xseq being sequence of Y such that
A61: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A62: xseq is convergent and
A63: tseq . x = lim xseq by A24;
A64: 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)).|| *
reconsider h1 = (vseq . m) - (vseq . k) as Lipschitzian LinearOperator of X,Y by Def7;
A65: xseq . k = (modetrans ((vseq . k),X,Y)) . x by A61;
vseq . m is Lipschitzian LinearOperator of X,Y by Def7;
then A66: modetrans ((vseq . m),X,Y) = vseq . m by Th28;
vseq . k is Lipschitzian LinearOperator of X,Y by Def7;
then A67: modetrans ((vseq . k),X,Y) = vseq . k by Th28;
xseq . m = (modetrans ((vseq . m),X,Y)) . x by A61;
then (xseq . m) - (xseq . k) = h1 . x by ;
hence ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * by Th31; :: thesis: verum
end;
A68: for m being Nat st m >= k holds
||.((xseq . n) - (xseq . m)).|| <= e *
proof
let m be Nat; :: thesis: ( m >= k implies ||.((xseq . n) - (xseq . m)).|| <= e * )
assume m >= k ; :: thesis: ||.((xseq . n) - (xseq . m)).|| <= e *
then A69: ||.((vseq . n) - (vseq . m)).|| < e by ;
A70: ||.((xseq . n) - (xseq . m)).|| <= ||.((vseq . n) - (vseq . m)).|| * by A64;
0 <= by CLVECT_1:105;
then ||.((vseq . n) - (vseq . m)).|| * <= e * by ;
hence ||.((xseq . n) - (xseq . m)).|| <= e * by ; :: thesis: verum
end;
||.((xseq . n) - (tseq . x)).|| <= e *
proof
deffunc H1( Nat) -> object = ||.((xseq . \$1) - (xseq . n)).||;
consider rseq being Real_Sequence such that
A71: for m being Nat holds rseq . m = H1(m) from SEQ_1:sch 1();
now :: thesis: for x being object st x in NAT holds
rseq . x = ||.(xseq - (xseq . n)).|| . x
let x be object ; :: thesis: ( x in NAT implies rseq . x = ||.(xseq - (xseq . n)).|| . x )
assume x in NAT ; :: thesis: rseq . x = ||.(xseq - (xseq . n)).|| . x
then reconsider k = x as Nat ;
thus rseq . x = ||.((xseq . k) - (xseq . n)).|| by A71
.= ||.((xseq - (xseq . n)) . k).|| by NORMSP_1:def 4
.= ||.(xseq - (xseq . n)).|| . x by NORMSP_0:def 4 ; :: thesis: verum
end;
then A72: rseq = ||.(xseq - (xseq . n)).|| by FUNCT_2:12;
A73: xseq - (xseq . n) is convergent by ;
lim (xseq - (xseq . n)) = (tseq . x) - (xseq . n) by ;
then A74: lim rseq = ||.((tseq . x) - (xseq . n)).|| by ;
for m being Nat st m >= k holds
rseq . m <= e *
proof
let m be Nat; :: thesis: ( m >= k implies rseq . m <= e * )
assume A75: m >= k ; :: thesis: rseq . m <= e *
rseq . m = ||.((xseq . m) - (xseq . n)).|| by A71
.= ||.((xseq . n) - (xseq . m)).|| by CLVECT_1:108 ;
hence rseq . m <= e * by ; :: thesis: verum
end;
then lim rseq <= e * by ;
hence ||.((xseq . n) - (tseq . x)).|| <= e * by ; :: thesis: verum
end;
hence ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * by A61; :: thesis: verum
end;
hence for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * ; :: thesis: verum
end;
hence for n being Nat st n >= k holds
for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * ; :: thesis: verum
end;
reconsider tseq = tseq as Lipschitzian LinearOperator of X,Y by A45;
reconsider tv = tseq as Point of by Def7;
A76: 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 A77: e > 0 ; :: thesis: ex k being Nat st
for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e

consider k being Nat such that
A78: for n being Nat st n >= k holds
for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * by ;
now :: thesis: for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e
set g1 = tseq;
let n be Nat; :: thesis: ( n >= k implies ||.((vseq . n) - tv).|| <= e )
assume A79: n >= k ; :: thesis: ||.((vseq . n) - tv).|| <= e
reconsider h1 = (vseq . n) - tv as Lipschitzian LinearOperator of X,Y by Def7;
set f1 = modetrans ((vseq . n),X,Y);
A80: now :: thesis: for t being VECTOR of X st <= 1 holds
||.(h1 . t).|| <= e
let t be VECTOR of X; :: thesis: ( <= 1 implies ||.(h1 . t).|| <= e )
assume ||.t.|| <= 1 ; :: thesis: ||.(h1 . t).|| <= e
then A81: e * <= e * 1 by ;
A82: ||.(((modetrans ((vseq . n),X,Y)) . t) - (tseq . t)).|| <= e * by ;
vseq . n is Lipschitzian LinearOperator of X,Y by Def7;
then modetrans ((vseq . n),X,Y) = vseq . n by Th28;
then ||.(h1 . t).|| = ||.(((modetrans ((vseq . n),X,Y)) . t) - (tseq . t)).|| by Th39;
hence ||.(h1 . t).|| <= e by ; :: thesis: verum
end;
A83: 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 VECTOR of X st
( r = ||.(h1 . t).|| & <= 1 ) ;
hence r <= e by A80; :: thesis: verum
end;
A84: ( ( for s being Real st s in PreNorms h1 holds
s <= e ) implies upper_bound (PreNorms h1) <= e ) by SEQ_4:45;
(BoundedLinearOperatorsNorm (X,Y)) . ((vseq . n) - tv) = upper_bound (PreNorms h1) by Th29;
hence ||.((vseq . n) - tv).|| <= e by ; :: thesis: verum
end;
hence ex k being Nat st
for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e ; :: thesis: verum
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 A85: e > 0 ; :: thesis: ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e

reconsider ee = e as Real ;
consider m being Nat such that
A86: for n being Nat st n >= m holds
||.((vseq . n) - tv).|| <= ee / 2 by ;
A87: e / 2 < e by ;
now :: thesis: for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e
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 A86;
hence ||.((vseq . n) - tv).|| < e by ; :: thesis: verum
end;
hence ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e ; :: thesis: verum
end;
hence vseq is convergent by CLVECT_1:def 15; :: thesis: verum