reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;

theorem Th19:
  for r be Real st 0<r & (for n being Nat holds seq.n=(1/(n+r))*x0) holds
  seq is convergent
proof
  let r be Real;
  assume that
A1: 0<r and
A2: for n being Nat holds seq.n=(1/(n+r))*x0;
  take g = 0.S;
  let p be Real;
  assume
A3: 0<p;
  ex pp be Real st pp > 0 & pp*||.x0.|| < p
  proof
    take pp=p/(||.x0.||+1);
A4: ||.x0.||+0 < ||.x0.||+1 & 0 <= ||.x0.|| by NORMSP_1:4,XREAL_1:8;
A5: ||.x0.||+1 > 0+0 by NORMSP_1:4,XREAL_1:8;
    then 0 < p/( ||.x0.||+1 ) by A3,XREAL_1:139;
    then pp* ||.x0.|| < pp*(||.x0.|| + 1) by A4,XREAL_1:97;
    hence thesis by A3,A5,XCMPLX_1:87;
  end;
  then consider pp be Real such that
A6: pp > 0 and
A7: pp*||.x0.|| < p;
  consider k1 be Nat such that
A8: pp"<k1 by SEQ_4:3;
  pp"+0<k1+r by A1,A8,XREAL_1:8;
  then 1/(k1+r)<1/pp" by A6,XREAL_1:76;
  then
A9: 1/(k1+r)<1*pp"" by XCMPLX_0:def 9;
   reconsider k1 as Element of NAT by ORDINAL1:def 12;
  take n=k1;
  let m be Nat;
  assume n<=m;
  then
A10: n+r<=m+r by XREAL_1:6;
A11: 0 <= ||.x0.|| by NORMSP_1:4;
  1/(m+r)<=1/(n+r) by A1,A10,XREAL_1:118;
  then 1/(m+r)<pp by A9,XXREAL_0:2;
  then
A12: (1/(m+r))*||.x0.|| <= pp* ||.x0.|| by A11,XREAL_1:64;
  ||.seq.m-g.|| = ||.(1/(m+r))*x0 - 0.S.|| by A2
    .= ||.(1/(m+r))*x0.|| by RLVECT_1:13
    .= |.1/(m+r).|*||.x0.|| by NORMSP_1:def 1
    .= (1/(m+r))*||.x0.|| by A1,ABSVALUE:def 1;
  hence thesis by A7,A12,XXREAL_0:2;
end;
