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$$\lim_{n\to\infty} \int_{D(n)} f dA = \int_{R ^2} f dA$$ exists,  We got $$\int_{D(n)} g dA \leq \int_{R ^2} f dA$$  (e) Since By the Monotone Convergence Theorem for sequences,  $\int_{D(n)} g dA$ fore has a limit increasing and bounded above is convergent  so   $\lim_{n\to\infty} \int_{D(n)} g dA = \lim_{n\to\infty} C_n$ exists