Prove the following Lebesgue's Monotone convergence theorem: Let 9n [0, 1] R be a sequence...

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Prove the following Lebesgue's Monotone convergence theorem: Let 9n [0, 1] → R be a sequence of non-negative functions such that each improper integral S9n(x) dx exists and is finite. Suppose that 0 ≤ 9n+1 ≤ In and 9n → 0 for each x € [0, 1]. Then So th lim n→∞ In(x) dx = 0. Prove the following Lebesgue's Monotone convergence theorem: Let 9n [0, 1] → R be a sequence of non-negative functions such that each improper integral S9n(x) dx exists and is finite. Suppose that 0 ≤ 9n+1 ≤ In and 9n → 0 for each x € [0, 1]. Then So th lim n→∞ In(x) dx = 0.