let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds s . n < 1 ) implies for n being Element of NAT holds (Partial_Sums s) . n < n + 1 )
assume A1: for n being Element of NAT holds s . n < 1 ; :: thesis: for n being Element of NAT holds (Partial_Sums s) . n < n + 1
defpred S₁[ Element of NAT ] means (Partial_Sums s) . $1 < $1 + 1;
(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1;
then A2: S₁[ 0 ] by A1;
A3: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A4: (Partial_Sums s) . n < n + 1 ; :: thesis: S₁[n + 1]
(Partial_Sums s) . (n + 1) = ((Partial_Sums s) . n) + (s . (n + 1)) by SERIES_1:def 1;
hence S₁[n + 1] by A1, A4, XREAL_1:10; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A3);
hence for n being Element of NAT holds (Partial_Sums s) . n < n + 1 ; :: thesis: verum