
theorem
  for s,t be Real_Sequence, g be Real st
  for n be Element of NAT holds t.n = |.s.n-g.| holds
  s is convergent & lim s = g iff t is convergent & lim t = 0
proof
  let s,t be Real_Sequence, g be Real;
  assume
A1: for n be Element of NAT holds t.n=|.s.n-g.|;
  hereby
    assume
A2: s is convergent & lim s = g;
A3: now
      let r be Real;
      assume 0 < r;
      then consider n be Nat such that
A4:   for m be Nat st n<=m holds |.s.m-g.| < r by A2,SEQ_2:def 7;
      take n;
      now
        let m be Nat;
A5:     m in NAT by ORDINAL1:def 12;
        assume n<=m;
        then |.|.s.m-g.|.| < r by A4;
        hence |.t.m-0 .| < r by A1,A5;
      end;
      hence for m be Nat st n<=m holds |.t.m-0 .|< r;
    end;
    hence t is convergent by SEQ_2:def 6;
    hence lim t = 0 by A3,SEQ_2:def 7;
  end;
  assume
A6: t is convergent & lim t = 0;
A7: now
    let r be Real;
    assume 0 < r;
    then consider n be Nat such that
A8: for m be Nat st n<=m holds |.t.m-0 .| < r by A6,SEQ_2:def 7;
    take n;
    now
      let m be Nat;
A9:      m in NAT by ORDINAL1:def 12;
      assume n<=m;
      then |.t.m-0 .| < r by A8;
      then |.|.s.m-g.|.| < r by A1,A9;
      hence |.s.m-g.| < r;
    end;
    hence for m be Nat st n <= m holds |.s.m-g.| < r;
  end;
  hence s is convergent by SEQ_2:def 6;
  hence thesis by A7,SEQ_2:def 7;
end;
