let X be RealNormSpace; :: thesis: for Sq being sequence of X st Sq is Cauchy_sequence_by_Norm holds
ex N being V174() sequence of NAT st
for i, j being Nat st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i)

let Sq be sequence of X; :: thesis: ( Sq is Cauchy_sequence_by_Norm implies ex N being V174() sequence of NAT st
for i, j being Nat st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i) )

assume A1: Sq is Cauchy_sequence_by_Norm ; :: thesis: ex N being V174() sequence of NAT st
for i, j being Nat st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i)

1 = 2 to_power () by POWER:24;
then consider N0 being Nat such that
A2: for j, i being Nat st j >= N0 & i >= N0 holds
||.((Sq . j) - (Sq . i)).|| < 2 to_power () by ;
reconsider N0 = N0 as Element of NAT by ORDINAL1:def 12;
defpred S1[ set , set , set ] means ex n, x, y being Nat st
( n = \$1 & x = \$2 & y = \$3 & ( ( for j being Nat st j >= x holds
||.((Sq . j) - (Sq . x)).|| < 2 to_power (- n) ) implies ( x < y & ( for j being Nat st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) ) ) );
A3: for n being Nat
for x being Element of NAT ex y being Element of NAT st S1[n,x,y]
proof
let n be Nat; :: thesis: for x being Element of NAT ex y being Element of NAT st S1[n,x,y]
let x be Element of NAT ; :: thesis: ex y being Element of NAT st S1[n,x,y]
now :: thesis: ( ( for j being Nat st j >= x holds
||.((Sq . j) - (Sq . x)).|| < 2 to_power (- n) ) implies ex y being Element of NAT st
( x < y & ( for j being Nat st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) ) )
assume for j being Nat st j >= x holds
||.((Sq . j) - (Sq . x)).|| < 2 to_power (- n) ; :: thesis: ex y being Element of NAT st
( x < y & ( for j being Nat st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) )

0 < 2 to_power (- (n + 1)) by POWER:34;
then consider N2 being Nat such that
A4: for j, i being Nat st j >= N2 & i >= N2 holds
||.((Sq . j) - (Sq . i)).|| < 2 to_power (- (n + 1)) by ;
set y = (max (x,N2)) + 1;
take y = (max (x,N2)) + 1; :: thesis: ( x < y & ( for j being Nat st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) )

x <= max (x,N2) by XXREAL_0:25;
hence x < y by NAT_1:13; :: thesis: for j being Nat st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1))

N2 <= max (x,N2) by XXREAL_0:25;
then A5: N2 < y by NAT_1:13;
thus for j being Nat st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) :: thesis: verum
proof
let j be Nat; :: thesis: ( j >= y implies ||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) )
assume j >= y ; :: thesis: ||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1))
then ( j >= N2 & y >= N2 ) by ;
hence ||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) by A4; :: thesis: verum
end;
end;
hence ex y being Element of NAT st S1[n,x,y] ; :: thesis: verum
end;
consider f being sequence of NAT such that
A6: ( f . 0 = N0 & ( for n being Nat holds S1[n,f . n,f . (n + 1)] ) ) from defpred S2[ Nat] means for j being Nat st j >= f . \$1 holds
||.((Sq . j) - (Sq . (f . \$1))).|| < 2 to_power (- \$1);
A7: S2[ 0 ] by A2, A6;
A8: now :: thesis: for i being Nat st S2[i] holds
S2[i + 1]
let i be Nat; :: thesis: ( S2[i] implies S2[i + 1] )
assume A9: S2[i] ; :: thesis: S2[i + 1]
ex n, x, y being Nat st
( n = i & x = f . i & y = f . (i + 1) & ( ( for j being Nat st j >= x holds
||.((Sq . j) - (Sq . x)).|| < 2 to_power (- n) ) implies ( x < y & ( for j being Nat st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) ) ) ) by A6;
hence S2[i + 1] by A9; :: thesis: verum
end;
A10: for i being Nat holds S2[i] from NAT_1:sch 2(A7, A8);
now :: thesis: for i being Nat holds f . i < f . (i + 1)
let i be Nat; :: thesis: f . i < f . (i + 1)
ex n, x, y being Nat st
( n = i & x = f . i & y = f . (i + 1) & ( ( for j being Nat st j >= x holds
||.((Sq . j) - (Sq . x)).|| < 2 to_power (- n) ) implies ( x < y & ( for j being Nat st j >= y holds
||.((Sq . j) - (Sq . y)).|| < 2 to_power (- (n + 1)) ) ) ) ) by A6;
hence f . i < f . (i + 1) by A10; :: thesis: verum
end;
then f is V174() ;
hence ex N being V174() sequence of NAT st
for i, j being Nat st j >= N . i holds
||.((Sq . j) - (Sq . (N . i))).|| < 2 to_power (- i) by A10; :: thesis: verum