Problem 1494 (difficulty: 8/10)

Show without Lebesgue theory that if \(\displaystyle f_n:[0,1]\to[0,1]\) is continuous for all \(\displaystyle n\) and \(\displaystyle f_n(x)\to 0\) for all \(\displaystyle x\in[0,1]\), then \(\displaystyle \int_0^1 f_n(x) \dx\to 0\) !

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