<P> Since all the summands are non-negative, the sum of the series, whether this sum is finite or infinite, cannot change if summation order does . For that reason, </P> <Dl> <Dd> ∑ i = 1 n c i ⋅ ∑ j = 1 ∞ μ (S j ∩ A i) = ∑ j = 1 ∞ ∑ i = 1 n c i ⋅ μ (S j ∩ A i) = ∑ j = 1 ∞ ∫ S j s d μ = ∑ j = 1 ∞ ν (S j), (\ displaystyle (\ begin (aligned) \ sum _ (i = 1) ^ (n) c_ (i) \ cdot \ sum _ (j = 1) ^ (\ infty) \ mu (S_ (j) \ cap A_ (i)) & = \ sum _ (j = 1) ^ (\ infty) \ sum _ (i = 1) ^ (n) c_ (i) \ cdot \ mu (S_ (j) \ cap A_ (i)) \ \ & = \ sum _ (j = 1) ^ (\ infty) \ int _ (S_ (j)) s \, d \ mu \ \ & = \ sum _ (j = 1) ^ (\ infty) \ nu (S_ (j)), \ end (aligned))) </Dd> </Dl> <Dd> ∑ i = 1 n c i ⋅ ∑ j = 1 ∞ μ (S j ∩ A i) = ∑ j = 1 ∞ ∑ i = 1 n c i ⋅ μ (S j ∩ A i) = ∑ j = 1 ∞ ∫ S j s d μ = ∑ j = 1 ∞ ν (S j), (\ displaystyle (\ begin (aligned) \ sum _ (i = 1) ^ (n) c_ (i) \ cdot \ sum _ (j = 1) ^ (\ infty) \ mu (S_ (j) \ cap A_ (i)) & = \ sum _ (j = 1) ^ (\ infty) \ sum _ (i = 1) ^ (n) c_ (i) \ cdot \ mu (S_ (j) \ cap A_ (i)) \ \ & = \ sum _ (j = 1) ^ (\ infty) \ int _ (S_ (j)) s \, d \ mu \ \ & = \ sum _ (j = 1) ^ (\ infty) \ nu (S_ (j)), \ end (aligned))) </Dd> <P> as required . </P>

A bounded increasing sequence of real numbers converges