reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;

theorem
  seq is convergent & (for n holds seq.n<=0) implies lim seq <= 0
proof
  assume that
A1: seq is convergent and
A2: for n holds seq.n<=0;
  set seq1 = -seq;
A3: now
    let n;
    seq1.n = -seq.n & seq.n<=0 by A2,SEQ_1:10;
    hence -(0 qua Real)<=seq1.n by XREAL_1:24;
  end;
  seq1 is convergent by A1,SEQ_2:9;
  then 0 <= lim seq1 by A3,SEQ_2:17;
  then -(0 qua Real) <= - lim seq by A1,SEQ_2:10;
  hence thesis by XREAL_1:24;
end;
