Choice of partition for Riemann integrationn

Let $$f:[0,1]\rightarrow \mathbb{R}$$ be a function. If $$P=\{0=a_0 < a_1 < \cdots < a_n=1\}$$ is any partition of $$[0,1]$$, then we define $$U(p,f)$$ and $$(L(p,f)$$, the upper and lower Riemann sums.

Suppose, we have the partition $$P_n=\{0,\frac{1}{n}, \frac{2}{n},\cdots,\frac{n}{n}=1\}$$ for $$[0,1]$$ and suppose we prove that $$|U(P_n,f)-L(P_n,f)|$$ tends to $$0$$ as $$n$$ tends to $$\infty$$. Can we conclude that $$f$$ is integrable on $$[0,1]$$?

I am considering this type of partition, for example, to check integrability of Thomae's function. (There was proof of the integrability but I was trying to pass through only specific types of partitions of $$[0,1]$$ and conclude the integrability; but I do not know, if this type of partition is sufficient for checking integrability.)

Yes, we can conclude that $$f$$ is Riemann integrable, according to an integrability criterion quoted in the accepted answer here.