theorem Th60:
  Rseq is P-convergent implies P-lim Rseq in
    lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).))
  proof
    assume
A1: Rseq is P-convergent;
    for m being non zero Nat ex n being Nat st for n1,n2 being Nat st
      n <= n1 & n <= n2 holds |. Rseq.(n1,n2) - P-lim Rseq .| < 1/m
    proof
      let m be non zero Nat;
      0/m < 1/m by XREAL_1:74;
      hence thesis by A1,DBLSEQ_1:def 2;
    end;
    hence P-lim Rseq in
    lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) by Th58;
  end;
