
theorem RNS11:
  for seq1 be sequence of RNS_Real
    st seq1 is Cauchy_sequence_by_Norm holds seq1 is convergent
proof
  let seq1 be sequence of RNS_Real;
  assume AS: seq1 is Cauchy_sequence_by_Norm;
  reconsider seq=seq1 as Real_Sequence;
  for s be Real st 0 < s ex n be Nat st
    for m be Nat st n <= m holds |.seq.m - seq.n.| < s
  proof
    let s be Real;
    assume 0 < s; then
    consider k be Nat such that
P1:   for n, m be Nat st n >= k & m >= k holds ||. seq1.n - seq1.m .|| < s
          by AS,RSSPACE3:8;
    take k;
    hereby let m be Nat;
      assume k <= m; then
P2:   ||.(seq1.m) - (seq1.k).|| < s by P1;
      (seq1.m) - (seq1.k) = (seq.m) - (seq.k) by RNS4;
      hence |.seq.m - seq.k.| < s by EUCLID:def 2,P2;
    end;
  end; then
  seq is convergent by SEQ_4:41;
  hence thesis by RNS8;
end;
