
theorem Th4:
  for s being Real_Sequence,S being sequence of RealSpace st s=S
  holds (s is convergent iff S is convergent) & (s is convergent implies lim s=
  lim S)
proof
  let s be Real_Sequence,S be sequence of RealSpace;
  assume
A1: s=S;
A2: s is convergent implies S is convergent
  proof
    assume s is convergent;
    then consider g being Real such that
A3: for p being Real st 0<p ex n being Nat st for m
    being Nat st n<=m holds |.s.m-g.| < p by SEQ_2:def 6;
    reconsider g0=g as Real;
    reconsider x0=g as Point of RealSpace by METRIC_1:def 13,XREAL_0:def 1;
    for r being Real st r>0 ex n being Nat st for m being
    Nat st n<=m holds dist(S.m,x0)<r
    proof
      let r be Real;
      assume r>0;
      then consider n0 being Nat such that
A4:   for m being Nat st n0<=m holds |.s.m-g.| < r by A3;
      for m being Nat st n0<=m holds dist(S.m,x0)<r
      proof
        let m be Nat;
        assume
A5:     n0<=m;
        reconsider t=s.m as Real;
        dist(S.m,x0)=real_dist.(t,g) by A1,METRIC_1:def 1,def 13
          .=|.t-g0.| by METRIC_1:def 12;
        hence thesis by A4,A5;
      end;
      hence thesis;
    end;
    hence thesis by TBSP_1:def 2;
  end;
  S is convergent implies s is convergent
  proof
    assume S is convergent;
    then consider x being Element of RealSpace such that
A6: for r being Real st r>0 ex n being Nat st for m being
    Nat st n<=m holds dist(S.m,x)<r by TBSP_1:def 2;
    reconsider g0=x as Real;
    for p being Real st 0<p ex n being Nat st for m
    being Nat st n<=m holds |.s.m-g0.| < p
    proof
      let p be Real;
      reconsider p0=p as Real;
      assume 0<p;
      then consider n0 being Nat such that
A7:   for m being Nat st n0<=m holds dist(S.m,x)<p0 by A6;
      for m being Nat st n0<=m holds |.s.m-g0.| < p
      proof
        let m be Nat;
        assume
A8:     n0<=m;
        reconsider t=s.m as Real;
        dist(S.m,x)=real_dist.(t,g0) by A1,METRIC_1:def 1,def 13
          .=|.t-g0.| by METRIC_1:def 12;
        hence thesis by A7,A8;
      end;
      hence thesis;
    end;
    hence thesis by SEQ_2:def 6;
  end;
  hence s is convergent iff S is convergent by A2;
  thus s is convergent implies lim s=lim S
  proof
    reconsider g0=lim S as Real;
    assume
A9: s is convergent;
    for r being Real st 0 < r ex m being Nat st for n
    being Nat st m <= n holds |.s.n-g0.| < r
    proof
      let r be Real;
      reconsider r0=r as Real;
      assume 0 < r;
      then consider m0 being Nat such that
A10:  for n being Nat st m0<=n holds dist(S.n,lim S)<r0 by A2,A9,
TBSP_1:def 3;
      for n being Nat st m0 <= n holds |.s.n -g0.| < r
      proof
        let n be Nat;
        assume
A11:    m0 <= n;
        dist(S.n,lim S)=real_dist.(s.n,g0) by A1,METRIC_1:def 1,def 13
          .=|.s.n-g0.| by METRIC_1:def 12;
        hence thesis by A10,A11;
      end;
      hence thesis;
    end;
    hence thesis by A9,SEQ_2:def 7;
  end;
end;
