
theorem Th8:
  for a,b being Real,s being Real_Sequence, S being
sequence of Closed-Interval-MSpace(a,b) st S=s & a<=b & s is convergent holds S
  is convergent & lim s=lim S
proof
  let a,b be Real,s be Real_Sequence, S be sequence of
  Closed-Interval-MSpace(a,b);
  assume that
A1: S=s and
A2: a<=b and
A3: s is convergent;
  reconsider S0=S as sequence of RealSpace by A2,Th6;
A4: S0 is convergent by A1,A3,Th4;
  hence S is convergent by A2,Th7;
  lim S0=lim S by A2,A4,Th7;
  hence thesis by A1,A3,Th4;
end;
