Prove or disprove: Suppose $f$ is bounded on the interval $[a,b]$ and for any $n\in\mathbb{N}$ there exist partitions $P_{n}$ and $Q_{n}$ such that $U(P_{n},f) \le \frac{1}{n}$ and $L(Q_{n},f) \ge -\frac{1}{n}$. Then $\int_{a}^{b} f$ exists and equals $0$.

Here's my attempt: $$-\frac{1}{n} \overset{(1)}{\leq} L(Q_{n},f) \overset{(2)}{\leq} \underline{\int_{a}^{b} f} \overset{(3)}{\leq} \overline{\int_{a}^{b} f} \overset{(4)}{\leq} U(P_{n},f) \overset{(5)}{\leq} \frac{1}{n}$$

The inequalities $(1),(5)$ are given and $(2),(3),(4)$ hold by definition of lower integral and upper integral. It follows by squeeze theorem that $\underline{\int_{a}^{b} f} = \overline{\int_{a}^{b} f} = 0$.

Is this proof okay?

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    $\begingroup$ After a while, I realized that $P_n$ and $Q_n$ are partitions. However, it would be nice if that were mentioned in the question. Otherwise, your proof looks good. $\endgroup$ – robjohn Jan 4 at 15:43
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    $\begingroup$ @robjohn i made the necessary changes $\endgroup$ – Ashish K Jan 4 at 15:45

Your proof is completely correct. Well done!

Note also that the following related statement is true:

A bounded function $f: [a,b] \to \mathbb{R}$ is Riemann-integrable if and only if there is a sequence of partitions $(P_n)$ such that

$$\lim_{n \to \infty} (U(f,P_n)-L(f,P_n)) = 0$$


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