set x0 = the Point of S;
deffunc H₁( Nat) -> Element of the carrier of S = (1 / (S + 1)) * the Point of S;
consider s1 being sequence of S such that
A1: for n being Element of NAT holds s1 . n = H₁(n) from FUNCT_2:sch 4();
A2: for n being Nat holds s1 . n = H₁(n)

proof

let n be Nat; :: thesis: s1 . n = H₁(n)
n in NAT by ORDINAL1:def 12;
hence s1 . n = H₁(n) by A1; :: thesis: verum

end;

take s1 ; :: thesis: s1 is 0. S -convergent
A3: lim s1 = 0. S by A2, Th20;
s1 is convergent by A2, Th19;
then s1 is 0. S -convergent by A3;
hence s1 is 0. S -convergent ; :: thesis: verum