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Limit of a sequence

Let f:[0, 1] \rightarrow (0, +\infty) be a continuous function and A be the set of all positive integers n such that there exists x_n such that

    \[\int_{x_n}^{1} f(t) \, \mathrm{d}t = \frac{1}{n}\]

Prove that \{x_n\}_{n \in A} is infinite and evaluate the limit

    \[\ell = \lim_{n \rightarrow +\infty} n \left( x_n -1 \right)\]

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