let n be Nat; :: thesis: for seq1 being sequence of (REAL-NS n)
for seq2 being sequence of (REAL-US n) st seq1 = seq2 & seq2 is Cauchy holds
seq1 is Cauchy_sequence_by_Norm
let seq1 be sequence of (REAL-NS n); :: thesis: for seq2 being sequence of (REAL-US n) st seq1 = seq2 & seq2 is Cauchy holds
seq1 is Cauchy_sequence_by_Norm
let seq2 be sequence of (REAL-US n); :: thesis: ( seq1 = seq2 & seq2 is Cauchy implies seq1 is Cauchy_sequence_by_Norm )
assume that
A1: seq1 = seq2 and
A2: seq2 is Cauchy ; :: thesis: seq1 is Cauchy_sequence_by_Norm
let r be Real; :: according to RSSPACE3:def 5 :: thesis: ( r <= 0 or ex b₁ being set st
for b₂, b₃ being set holds
( not b₁ <= b₂ or not b₁ <= b₃ or not r <= dist ((seq1 . b₂),(seq1 . b₃)) ) )
assume r > 0 ; :: thesis: ex b₁ being set st
for b₂, b₃ being set holds
( not b₁ <= b₂ or not b₁ <= b₃ or not r <= dist ((seq1 . b₂),(seq1 . b₃)) )
then consider k being Nat such that
A3: for m1, m2 being Nat st m1 >= k & m2 >= k holds
dist ((seq2 . m1),(seq2 . m2)) < r by A2;
take k ; :: thesis: for b₁, b₂ being set holds
( not k <= b₁ or not k <= b₂ or not r <= dist ((seq1 . b₁),(seq1 . b₂)) )
let m1, m2 be Nat; :: thesis: ( not k <= m1 or not k <= m2 or not r <= dist ((seq1 . m1),(seq1 . m2)) )
reconsider p = (seq2 . m1) - (seq2 . m2) as Element of REAL n by Def6;
- (seq1 . m2) = - (seq2 . m2) by A1, Th13;
then A4: p = (seq1 . m1) - (seq1 . m2) by A1, Th13;
assume ( m1 >= k & m2 >= k ) ; :: thesis: not r <= dist ((seq1 . m1),(seq1 . m2))
then A5: dist ((seq2 . m1),(seq2 . m2)) < r by A3;
||.((seq2 . m1) - (seq2 . m2)).|| = sqrt ((Euclid_scalar n) . (p,p)) by Def6
.= sqrt (Sum (mlt (p,p))) by Def5
.= sqrt |(p,p)| by RVSUM_1:def 16
.= |.p.| by EUCLID_2:5
.= ||.((seq1 . m1) - (seq1 . m2)).|| by A4, Th1 ;
hence not r <= dist ((seq1 . m1),(seq1 . m2)) by A5; :: thesis: verum