reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th41:
  for X be RealNormSpace for seq be sequence of X st seq is
  convergent holds ||.seq.|| is convergent & lim ||.seq.|| = ||.lim seq.||
proof
  let X be RealNormSpace;
  let seq be sequence of X such that
A1: seq is convergent;
A2: now
    let e1 be Real such that
A3: e1 >0;
    reconsider e =e1 as Real;
    consider k be Nat such that
A4: for n be Nat st n >= k holds ||.seq.n - lim seq.|| < e
    by A1,A3,NORMSP_1:def 7;
     reconsider k as Nat;
    take k;
    now
      let m be Nat;
      assume m >= k;
      then
A5:   ||.seq.m - lim seq.|| <e by A4;
      ||.seq.m.||= ||.seq.||.m by NORMSP_0:def 4;
      then |. ||.seq.||.m - ||.lim seq.|| .| <= ||.seq.m - lim seq.|| by
NORMSP_1:9;
      hence |. ||.seq.||.m - ||.lim seq.|| .| <e1 by A5,XXREAL_0:2;
    end;
    hence for m be Nat st m >= k holds |.||.seq.||.m - ||.lim seq
    .|| .| < e1;
  end;
  then ||.seq.|| is convergent by SEQ_2:def 6;
  hence thesis by A2,SEQ_2:def 7;
end;
