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 Th20:
  for r be Real st 0<r &
   (for n being Nat holds seq.n=(1/(n+r))*x0 ) holds lim seq=0.S
proof
  let r be Real;
  assume that
A1: 0<r and
A2: for n being Nat holds seq.n=1/(n+r)*x0;
A3: now
    let p;
A4: 0 <= ||.x0.|| by NORMSP_1:4;
    assume
A5: 0<p;
    ex pp be Real st pp > 0 & pp*||.x0.|| < p
    proof
      take pp=p/(||.x0.||+1);
A6:   ||.x0.||+0 < ||.x0.||+1 & 0 <= ||.x0.|| by NORMSP_1:4,XREAL_1:8;
A7:   ||.x0.||+1 > 0+0 by NORMSP_1:4,XREAL_1:8;
      then 0 < p/(||.x0.||+1) by A5,XREAL_1:139;
      then pp* ||.x0.|| < pp*(||.x0.|| + 1) by A6,XREAL_1:97;
      hence thesis by A5,A7,XCMPLX_1:87;
    end;
    then consider pp be Real such that
A8: pp > 0 and
A9: pp*||.x0.|| < p;
    consider k1 be Nat such that
A10: pp"<k1 by SEQ_4:3;
    pp"+0<k1+r by A1,A10,XREAL_1:8;
    then 1/(k1+r)<1/pp" by A8,XREAL_1:76;
    then
A11: 1/(k1+r)<1*pp"" by XCMPLX_0:def 9;
     reconsider n=k1 as Nat;
    take n;
    let m be Nat;
    assume n<=m;
    then
A12: n+r<=m+r by XREAL_1:6;
    1/(m+r)<=1/(n+r) by A1,A12,XREAL_1:118;
    then 1/(m+r)<pp by A11,XXREAL_0:2;
    then
A13: (1/(m+r))*||.x0.|| <= pp* ||.x0.|| by A4,XREAL_1:64;
    ||.seq.m-0.S.|| = ||.(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 ||.seq.m-0.S.||<p by A9,A13,XXREAL_0:2;
  end;
  seq is convergent by A1,A2,Th19;
  hence thesis by A3,NORMSP_1:def 7;
end;
