Theorem 1: Suppose f(m, n) is a double sequence on R such that f(m, n) > 0. Then, E f(m, n) converges if and only if the set of partial sum is bounded. Proof. The detailed is left as exercise. Just only apply monotone convergence theorem. Theorem 2: If E f(m,n) converges absolutely, then Ef(m,n) converges. Proof. Left as exercise.

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