If there is $\{P_n\}$ of partitions with $\lim_{n\to\infty}[U(f,P_n) - L(f,P_n)]=0,$then $f$ is Riemann integrable.

Question

The following question is taken from Real Analysis by Royden, Chapter $4,$ question $5:$

Let $f$ be a bounded function on $[a,b].$ Suppose there is a sequence $\{P_n\}$ of partitions of $[a,b]$ for which $\lim_{n\to\infty}[U(f,P_n) - L(f,P_n)]=0.$ Show that $f$ is Riemann integrable over $[a,b].$

My attempt:

Let $\varepsilon>0$ be given. Then there exists a partition $P_N$ of $[a,b]$ such that $$U(f,P_N)-L(f,P_N) < \varepsilon.$$ Since $\varepsilon>0$ is arbitrary, therefore $$U(f,P_N)\leq L(f,P_N) \leq \underline{\int}_{a}^b f.$$ Similarly, $$\overline{\int}_a^b f \leq U(f,P_N) \leq L(f,P_N) \leq \underline{\int}_a^b f.$$ Therefore, $f$ is Riemann integrable on $[a,b].$

Is my proof correct?

Answer 1

No. $P_N$ depends on $\varepsilon$ !

We have $\overline{\int}_a^b f \le U(f;P_n) < L(f;P_n)+\varepsilon \le \underline{\int}_a^b f+\varepsilon $.

Hence $\overline{\int}_a^b f < \underline{\int}_a^b f+\varepsilon $.

Now let $ \varepsilon \to 0+$.