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
  for a be 0-convergent non-zero Real_Sequence
    for z be Point of S st z <> 0.S
  holds a*z is non-zero (0.S)-convergent
proof
  let a be 0-convergent non-zero Real_Sequence;
  let z be Point of S such that
A1: z <> 0.S;
  now
    let n be Nat;
A2: a.n <> 0 by SEQ_1:5;
    assume (a*z).n = 0.S;
    then a.n*z = 0.S by Def3;
    hence contradiction by A1,A2,RLVECT_1:11;
  end;
  hence a*z is non-zero by Th7;
A3: now
    let p;
    assume
A4: 0<p;
    ex pp be Real st pp > 0 & pp*||.z.|| < p
    proof
      take pp=p/(||.z.||+1);
A5:   ||.z.||+0 < ||.z.||+1 & 0 <= ||.z.|| by NORMSP_1:4,XREAL_1:8;
A6:   ||.z.||+1 > 0+0 by NORMSP_1:4,XREAL_1:8;
      then 0 < p/(||.z.||+1) by A4,XREAL_1:139;
      then pp* ||.z.|| < pp*(||.z.|| + 1) by A5,XREAL_1:97;
      hence thesis by A4,A6,XCMPLX_1:87;
    end;
    then consider pp be Real such that
A7: pp > 0 and
A8: pp*||.z.|| < p;
    a is convergent & lim a =0;
    then consider n be Nat such that
A9: for m be Nat st n <= m holds |.a.m- 0 .| < pp by A7,SEQ_2:def 7;
    reconsider n as Nat;
    take n;
    let m be Nat;
    assume n<=m;
    then
A10: |.a.m-0 .| < pp by A9;
A11: ||.(a*z).m-0.S.|| = ||.a.m*z - 0.S.|| by Def3
      .= ||.a.m*z.|| by RLVECT_1:13
      .= |.a.m.|*||.z.|| by NORMSP_1:def 1;
    0 <= ||.z.|| by NORMSP_1:4;
    then |.a.m.|*||.z.|| <= pp* ||.z.|| by A10,XREAL_1:64;
    hence ||.(a*z).m-0.S.||<p by A8,A11,XXREAL_0:2;
  end;
  hence a*z is convergent;
  hence thesis by A3,NORMSP_1:def 7;
end;
