reserve x for object;
reserve x0,r,r1,r2,g,g1,g2,p,y0 for Real;
reserve n,m,k,l for Element of NAT;
reserve a,b,d for Real_Sequence;
reserve h,h1,h2 for non-zero 0-convergent Real_Sequence;
reserve c,c1 for constant Real_Sequence;
reserve A for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve L for LinearFunc;
reserve R for RestFunc;

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;
