let X be non empty set ; :: thesis: for F being Functional_Sequence of X,ExtREAL
for n being Nat st ( for m being Nat holds F . m is without-infty ) holds
(Partial_Sums F) . n is without-infty
let F be Functional_Sequence of X,ExtREAL; :: thesis: for n being Nat st ( for m being Nat holds F . m is without-infty ) holds
(Partial_Sums F) . n is without-infty
let n be Nat; :: thesis: ( ( for m being Nat holds F . m is without-infty ) implies (Partial_Sums F) . n is without-infty )
defpred S₁[ Nat] means (Partial_Sums F) . $1 is without-infty;
assume A1: for m being Nat holds F . m is without-infty ; :: thesis: (Partial_Sums F) . n is without-infty
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
A4: F . (k + 1) is without-infty by A1;
(Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4;
hence S₁[k + 1] by A3, A4, Th3; :: thesis: verum

end;

(Partial_Sums F) . 0 = F . 0 by Def4;
then A5: S₁[ 0 ] by A1;
for k being Nat holds S₁[k] from NAT_1:sch 2(A5, A2);
hence (Partial_Sums F) . n is without-infty ; :: thesis: verum