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Prove or disprove: If lim f(x) = 0 and lim In = A then ∫ f(x) converges to A

The following function f is defined for all x \geq 1, and n is a positive integer. Prove or provide a counterexample to the following statement.

Assume

    \[ \lim_{x \to \infty} f(x) = 0 \qquad \text{and} \qquad \lim_{n \to \infty} I_n = A. \]

Then

    \[ \int_1^{\infty} f(x) \, dx = A. \]


Incomplete.

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