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A sequence of integrals, similar to a Putnam problem

Source: Romanian District Olympiad 2016, Grade XII, Problem 4

October 5, 2018
functioncalculusintegrationPutnam

Problem Statement

Let f:[0,1]⟶[0,1] f:[0,1]\longrightarrow [0,1] be a nondecreasing function. Prove that the sequence (∫011+fn(x)1+f1+n(x))n≥1 \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} is convergent and calculate its limit.
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