let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n > 0 ) implies for n being Nat holds (Partial_Sums s) . n > 0 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 > 0 ;
assume A1: for n being Nat holds s . n > 0 ; :: thesis: for n being Nat holds (Partial_Sums s) . n > 0
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
A3: (Partial_Sums s) . (n + 1) = ((Partial_Sums s) . n) + (s . (n + 1)) by SERIES_1:def 1;
assume (Partial_Sums s) . n > 0 ; :: thesis: S₁[n + 1]
hence S₁[n + 1] by A1, A3, XREAL_1:34; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1;
then A4: S₁[ 0 ] by A1;
for n being Nat holds S₁[n] from NAT_1:sch 2(A4, A2);
hence for n being Nat holds (Partial_Sums s) . n > 0 ; :: thesis: verum