let X be RealUnitarySpace; :: thesis: for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds
seq1 - seq2 is Cauchy
let seq1, seq2 be sequence of X; :: thesis: ( seq1 is Cauchy & seq2 is Cauchy implies seq1 - seq2 is Cauchy )
assume that
A1: seq1 is Cauchy and
A2: seq2 is Cauchy ; :: thesis: seq1 - seq2 is Cauchy
let r be Real; :: according to BHSP_3:def 1 :: thesis: ( r > 0 implies ex k being Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
dist ((seq1 - seq2) . n),((seq1 - seq2) . m) < r )
assume r > 0 ; :: thesis: ex k being Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
dist ((seq1 - seq2) . n),((seq1 - seq2) . m) < r
then A3: r / 2 > 0 by XREAL_1:217;
then consider m1 being Element of NAT such that
A4: for n, m being Element of NAT st n >= m1 & m >= m1 holds
dist (seq1 . n),(seq1 . m) < r / 2 by A1, Def1;
consider m2 being Element of NAT such that
A5: for n, m being Element of NAT st n >= m2 & m >= m2 holds
dist (seq2 . n),(seq2 . m) < r / 2 by A2, A3, Def1;
take k = m1 + m2; :: thesis: for n, m being Element of NAT st n >= k & m >= k holds
dist ((seq1 - seq2) . n),((seq1 - seq2) . m) < r
let n, m be Element of NAT ; :: thesis: ( n >= k & m >= k implies dist ((seq1 - seq2) . n),((seq1 - seq2) . m) < r )
assume A6: ( n >= k & m >= k ) ; :: thesis: dist ((seq1 - seq2) . n),((seq1 - seq2) . m) < r
k >= m2 by NAT_1:12;
then ( n >= m2 & m >= m2 ) by A6, XXREAL_0:2;
then A7: dist (seq2 . n),(seq2 . m) < r / 2 by A5;
dist ((seq1 - seq2) . n),((seq1 - seq2) . m) = dist ((seq1 . n) - (seq2 . n)),((seq1 - seq2) . m) by NORMSP_1:def 6
.= dist ((seq1 . n) - (seq2 . n)),((seq1 . m) - (seq2 . m)) by NORMSP_1:def 6 ;
then A8: dist ((seq1 - seq2) . n),((seq1 - seq2) . m) <= (dist (seq1 . n),(seq1 . m)) + (dist (seq2 . n),(seq2 . m)) by BHSP_1:48;
m1 + m2 >= m1 by NAT_1:12;
then ( n >= m1 & m >= m1 ) by A6, XXREAL_0:2;
then dist (seq1 . n),(seq1 . m) < r / 2 by A4;
then (dist (seq1 . n),(seq1 . m)) + (dist (seq2 . n),(seq2 . m)) < (r / 2) + (r / 2) by A7, XREAL_1:10;
hence dist ((seq1 - seq2) . n),((seq1 - seq2) . m) < r by A8, XXREAL_0:2; :: thesis: verum