let X be ComplexUnitarySpace; :: thesis: for seq1, seq2 being sequence of X st seq1 is convergent & seq1 is_compared_to seq2 holds
seq2 is convergent
let seq1, seq2 be sequence of X; :: thesis: ( seq1 is convergent & seq1 is_compared_to seq2 implies seq2 is convergent )
assume that
A1: seq1 is convergent and
A2: seq1 is_compared_to seq2 ; :: thesis: seq2 is convergent

now :: thesis: for r being Real st r > 0 holds
ex m being set st
for n being Nat st n >= m holds
dist ((seq2 . n),(lim seq1)) < r

let r be Real; :: thesis: ( r > 0 implies ex m being set st
for n being Nat st n >= m holds
dist ((seq2 . n),(lim seq1)) < r )
assume r > 0 ; :: thesis: ex m being set st
for n being Nat st n >= m holds
dist ((seq2 . n),(lim seq1)) < r
then A3: r / 2 > 0 by XREAL_1:215;
then consider m1 being Nat such that
A4: for n being Nat st n >= m1 holds
dist ((seq1 . n),(lim seq1)) < r / 2 by A1, Def2;
consider m2 being Nat such that
A5: for n being Nat st n >= m2 holds
dist ((seq1 . n),(seq2 . n)) < r / 2 by A2, A3;
take m = m1 + m2; :: thesis: for n being Nat st n >= m holds
dist ((seq2 . n),(lim seq1)) < r
let n be Nat; :: thesis: ( n >= m implies dist ((seq2 . n),(lim seq1)) < r )
assume A6: n >= m ; :: thesis: dist ((seq2 . n),(lim seq1)) < r
m >= m2 by NAT_1:12;
then n >= m2 by A6, XXREAL_0:2;
then A7: dist ((seq1 . n),(seq2 . n)) < r / 2 by A5;
A8: dist ((seq2 . n),(lim seq1)) <= (dist ((seq2 . n),(seq1 . n))) + (dist ((seq1 . n),(lim seq1))) by CSSPACE:51;
m1 + m2 >= m1 by NAT_1:12;
then n >= m1 by A6, XXREAL_0:2;
then dist ((seq1 . n),(lim seq1)) < r / 2 by A4;
then (dist ((seq2 . n),(seq1 . n))) + (dist ((seq1 . n),(lim seq1))) < (r / 2) + (r / 2) by A7, XREAL_1:8;
hence dist ((seq2 . n),(lim seq1)) < r by A8, XXREAL_0:2; :: thesis: verum

end;

hence seq2 is convergent ; :: thesis: verum