# Prove or disprove if $\int_{a}^{b} f$ exists and equals 0

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?

• 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. – robjohn Jan 4 at 15:43
• @robjohn i made the necessary changes – Ashish K Jan 4 at 15:45

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