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# Romanian District Mathematical Olympiad, 12th grade, 2001, problem 4

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Tags:competition, Contest, Convergence, Integrals, Integration, Lebesgue, Limit, Limits, math, Mathematics, Maths, Measure, Measure theory, Monotone, Monotone convergence theorem, Olympiad, RMO, RMO 2001, romania, Romania 2001, Romanian District, Romanian Mathematical Olympiad
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f'(x) is negative 1/1+x – 1 = -x/1+x . you accidentally took the the function to be ln(1+x)-x instead of x-ln(1+x)

As it was noticed by @jawinfernandes9355, I made a mistake defining function 𝑓, I just wrote it the other way around. It should be 𝑓(𝑥) = 𝑥 – ln(𝑥 + 1) — then its derivative is positive over the set of positive real numbers, so this function is strictly increasing. Moreover, 𝑓(0) = 0, so ln(𝑥 + 1) < 𝑥 for 𝑥 > 0.

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