let X be RealUnitarySpace; :: thesis: for seq1, seq2 being sequence of X holds
( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r )
let seq1, seq2 be sequence of X; :: thesis: ( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r )
thus ( seq1 is_compared_to seq2 implies for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r ) :: thesis: ( ( for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r ) implies seq1 is_compared_to seq2 )

proof

assume A1: seq1 is_compared_to seq2 ; :: thesis: for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r
let r be Real; :: thesis: ( r > 0 implies ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r )
assume r > 0 ; :: thesis: ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r
then consider m1 being Nat such that
A2: for n being Nat st n >= m1 holds
dist ((seq1 . n),(seq2 . n)) < r by A1;
take m = m1; :: thesis: for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r
let n be Nat; :: thesis: ( n >= m implies ||.((seq1 . n) - (seq2 . n)).|| < r )
assume n >= m ; :: thesis: ||.((seq1 . n) - (seq2 . n)).|| < r
then dist ((seq1 . n),(seq2 . n)) < r by A2;
hence ||.((seq1 . n) - (seq2 . n)).|| < r by BHSP_1:def 5; :: thesis: verum

end;

( ( for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r ) implies seq1 is_compared_to seq2 )

proof

assume A3: for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r ; :: thesis: seq1 is_compared_to seq2
let r be Real; :: according to BHSP_3:def 2 :: thesis: ( r > 0 implies ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r )
assume r > 0 ; :: thesis: ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r
then consider m1 being Nat such that
A4: for n being Nat st n >= m1 holds
||.((seq1 . n) - (seq2 . n)).|| < r by A3;
take m = m1; :: thesis: for n being Nat st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r
let n be Nat; :: thesis: ( n >= m implies dist ((seq1 . n),(seq2 . n)) < r )
assume n >= m ; :: thesis: dist ((seq1 . n),(seq2 . n)) < r
then ||.((seq1 . n) - (seq2 . n)).|| < r by A4;
hence dist ((seq1 . n),(seq2 . n)) < r by BHSP_1:def 5; :: thesis: verum

end;

hence ( ( for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r ) implies seq1 is_compared_to seq2 ) ; :: thesis: verum