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