Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions?

Question

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions?

I know that by the definition of an Archmedean sequence of partitions, we have

$$\lim_{k\to\infty}[U(f, P_{k}) - L(f, P_{k})] = 0. $$

Also by Archimedean Riemann, we get

$$\lim_{k\to\infty} L(f, P_{k}) = \int_{I} f = \lim_{k\to\infty} U(f,P_{k}). $$

So it's nowhere stated, but I can't come up with a counterexample. Any help is appreciated.