reserve x,X for set;
reserve x0,r1,r2,g,g1,g2,p,s for Real;
reserve r for Real;
reserve n,m for Nat;
reserve a,b,d for Real_Sequence;
reserve f for PartFunc of REAL,REAL;

theorem Th5:
  for x0 be Real holds (for n holds a.n = x0 - p/(n+1))
  implies a is convergent & lim a = x0
proof
  let x0 be Real;
  deffunc F(Nat) = p/($1+1);
  consider d such that
A1: for n holds d.n = F(n) from SEQ_1:sch 1;
  set b = seq_const x0;
  assume
A2: for n holds a.n = x0 - p/(n+1);
  now
    let n be Element of NAT;
    thus (b - d).n = b.n - d.n by VALUED_1:15
      .= x0 - d.n by SEQ_1:57
      .= x0 - p/(n+1) by A1
      .=a.n by A2;
  end;
  then
A3: a = b - d by FUNCT_2:63;
A4: d is convergent by A1,SEQ_4:31;
  hence a is convergent by A3;
A5: lim b = b.0 by SEQ_4:26
    .= x0 by SEQ_1:57;
  lim d = 0 by A1,SEQ_4:31;
  hence lim a = x0 - 0 by A4,A5,A3,SEQ_2:12
    .= x0;
end;
