theorem Th6:
  a is subsequence of h implies a is 0-convergent non-zero Real_Sequence
proof
  assume
A1: a is subsequence of h;
  then consider I be increasing sequence of NAT such that
A2: a = h*I by VALUED_0:def 17;
  not ex n being Nat st a.n = 0 by A2,SEQ_1:5;
  then
A3: a is non-zero by SEQ_1:5;
A4: a is convergent by A1,SEQ_4:16;
  lim h = 0;
  then lim a = 0 by A1,SEQ_4:17;
  hence thesis by A4,A3,FDIFF_1:def 1;
end;
