
theorem
  for S1 be sequence of RealSpace, seq be Real_Sequence
    st S1 = seq & seq is convergent
  holds S1 is convergent & lim S1 = lim seq
  proof
    let S1 be sequence of RealSpace, seq be Real_Sequence;
    assume
A1: S1 = seq & seq is convergent;
    reconsider g1 = lim seq as Point of RealSpace by XREAL_0:def 1;
    for p be Real st 0 < p
    ex n be Nat st for m be Nat st n <= m holds |.seq.m - lim seq.| < p
      by A1,SEQ_2:def 7; then
A2: for p be Real st 0 < p
      ex n be Nat st for m be Nat st n <= m holds dist(S1.m,g1) < p
        by A1,Th23;
    hence S1 is convergent;
    hence lim S1 = lim seq by A2,TBSP_1:def 3;
  end;
