reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

theorem
  abs seq is convergent & lim abs seq=0 implies seq is convergent & lim seq=0
proof
  assume that
A1: abs seq is convergent and
A2: lim abs seq=0;
A3: now
    let n;
    -|.seq.n.|<=seq.n by ABSVALUE:4;
    then
A4: -((abs seq).n)<=seq.n by SEQ_1:12;
    seq.n<=|.seq.n.| by ABSVALUE:4;
    hence (-abs seq).n<=seq.n & seq.n<=(abs seq).n by A4,SEQ_1:10,12;
  end;
A5: -abs seq is convergent & lim(-abs seq)=-lim abs seq by A1,SEQ_2:10;
  hence seq is convergent by A1,A2,A3,SEQ_2:19;
  thus thesis by A1,A2,A5,A3,SEQ_2:20;
end;
