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 Th23:
  seq is convergent & lim seq<>0 implies ex k st (seq ^\k) is non-zero
proof
  assume that
A1: seq is convergent and
A2: lim seq<>0;
  consider n1 such that
A3: for m st n1<=m holds |.lim seq.|/2<|.seq.m.| by A1,A2,SEQ_2:16;
  take k=n1;
  now
    let n;
    (0 qua Nat)+k<=n+k by XREAL_1:7;
    then |.lim seq.|/2<|.(seq.(n+k)).| by A3;
    then
A4: |.lim seq.|/2<|.((seq ^\k).n).| by NAT_1:def 3;
    0<|.lim seq.| by A2,COMPLEX1:47;
    then 0/2<|.lim seq.|/2;
    hence (seq ^\k).n<>0 by A4,ABSVALUE:2;
  end;
  hence thesis by SEQ_1:5;
end;
