Absolute convergence proof

Fermat

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Let R be a sigma algebra and let $f$ be a real value function on R such that for a sequence
($A_{n}$) of disjoint members of R, we have that the sum of $f$($A_{n}$) over all n is equal to the image of the countable union under $f$. Prove that the sum of $f$($A_{n}$) is in fact absolutely convergent.