reserve n, m for Element of NAT;
reserve z for Complex;

theorem Th3:
  for seq be Real_Sequence holds seq is convergent & lim seq = 0
  iff abs seq is convergent & lim abs seq = 0
proof
  let seq be Real_Sequence;
  thus seq is convergent & lim seq = 0 implies abs seq is convergent & lim abs
  seq = 0
  proof
    assume that
A1: seq is convergent and
A2: lim seq = 0;
    lim abs seq = |.0 .| by A1,A2,SEQ_4:14
      .= 0 by ABSVALUE:2;
    hence thesis by A1;
  end;
  thus thesis by SEQ_4:15;
end;
