let seq1 be sequence of X; :: thesis: for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),(seq1 . n)) < r
let r be Real; :: thesis: ( r > 0 implies ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),(seq1 . n)) < r )
assume A1: r > 0 ; :: thesis: ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),(seq1 . n)) < r
take m = 0 ; :: thesis: for n being Nat st n >= m holds
dist ((seq1 . n),(seq1 . n)) < r
let n be Nat; :: thesis: ( n >= m implies dist ((seq1 . n),(seq1 . n)) < r )
assume n >= m ; :: thesis: dist ((seq1 . n),(seq1 . n)) < r
thus dist ((seq1 . n),(seq1 . n)) < r by A1, BHSP_1:34; :: thesis: verum