reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);
reserve v,u for VECTOR of RLSp_AlmostZeroLpFunct(M,k);
reserve x for Point of Pre-Lp-Space(M,k);
reserve x,y for Point of Lp-Space(M,k);

theorem
for X be RealNormSpace, Sq be sequence of X, Sq0 be Point of X st
 ||.Sq -Sq0.|| is convergent & lim ||.Sq -Sq0.|| =0
holds Sq is convergent & lim Sq = Sq0
proof
   let X be RealNormSpace, Sq be sequence of X, Sq0 be Point of X;
   assume
A1: ||.Sq -Sq0.|| is convergent & lim ||.Sq -Sq0.|| = 0;
A2:for p be Real st 0 < p ex n be Nat st
     for m be Nat st n <= m holds ||. Sq.m - Sq0 .|| < p
   proof
    let p be Real;
    assume 0 < p; then
    consider n such that
A3:  for m st n<=m holds |. (||.Sq - Sq0.||).m - 0 qua Complex .| < p
          by A1,SEQ_2:def 7;
    take n;
    hereby let m be Nat;
     assume n<=m; then
     |.(||.Sq -Sq0.||).m-0 .| < p by A3; then
     |.||. (Sq -Sq0).m .||.| < p by NORMSP_0:def 4; then
     |.||.Sq.m -Sq0.||.| < p by NORMSP_1:def 4;
     hence ||.Sq.m -Sq0.|| < p by ABSVALUE:def 1;
    end;
   end;
   hence Sq is convergent by NORMSP_1:def 6;
   hence lim Sq =Sq0 by A2,NORMSP_1:def 7;
end;
