let rseq be Real_Sequence; :: thesis: for p being Real st ( for n being Nat holds rseq . n <= p ) holds
for n being Nat holds (Partial_Sums rseq) . n <= p * (n + 1)
let p be Real; :: thesis: ( ( for n being Nat holds rseq . n <= p ) implies for n being Nat holds (Partial_Sums rseq) . n <= p * (n + 1) )
defpred S₁[ Nat] means (Partial_Sums rseq) . $1 <= p * ($1 + 1);
assume A1: for n being Nat holds rseq . n <= p ; :: thesis: for n being Nat holds (Partial_Sums rseq) . n <= p * (n + 1)

A2: now :: thesis: for n being Nat st S₁[n] holds
S₁[n + 1]

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A3: S₁[n] ; :: thesis: S₁[n + 1]
rseq . (n + 1) <= p by A1;
then A4: (p * (n + 1)) + (rseq . (n + 1)) <= (p * (n + 1)) + p by XREAL_1:6;
(Partial_Sums rseq) . (n + 1) = ((Partial_Sums rseq) . n) + (rseq . (n + 1)) by SERIES_1:def 1;
then (Partial_Sums rseq) . (n + 1) <= (p * (n + 1)) + (rseq . (n + 1)) by A3, XREAL_1:6;
hence S₁[n + 1] by A4, XXREAL_0:2; :: thesis: verum

end;

(Partial_Sums rseq) . 0 = rseq . 0 by SERIES_1:def 1;
then A5: S₁[ 0 ] by A1;
thus for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A2); :: thesis: verum