Why author has not specified what actually happens when $n\to \infty$ for the sequence $P_n$? In my book the author is giving the rigorous definition of integrals and here is what he writes:

Let $f: [a,b] \mapsto \mathbb R $ be a bounded function. Then $f$ is integrable on $[a,b]$ if and only if there is a sequence $\{P_n\}$ of partitions of the interval $[a,b]$ such that $$\lim_{n\to \infty} \left[ U(f, P_n) - L(f,P_n)\right] =0$$.

But what I don’t understand is that the author doesn’t ever say that as $n$ gets larger and larger $P_n$ gets finer and finer. If he doesn’t define how $P_n$ depends on $n$ then what’s the meaning of defining something like
$$
\lim_{n\to \infty} \left[
         U(f,P_n) -L(f,P_n) \right]
$$
From what I think he should say that $n$ suggests the number of points in the partition $P_n$, because then it would be easier to see why upper and lower sum will converge (upper sum will decrease, lower sum will increase).
But when I asked it to someone, they replied that $$\lim_{n\to \infty} [U(f,P_n)-L(f,P_n)]$$ is perfectly fine and they used the concept of limit point/accumulation point to explain that it’s valid, but I couldn’t convince myself with that.
Please express yourself about how you think about it.
 A: It really doesn't matter if $\{P_n\}$ becomes finer or contains more points as $n$ increases. Indeed, let $\mathcal{P}$ denote the set of all partitions of $[a, b]$ and consider two sets
\begin{align*}
A &= \{ L(f, P) : P \in \mathcal{P} \}, &
B &= \{ U(f, P) : P \in \mathcal{P} \}.
\end{align*}
Then we have the following observations:

*

*The inequality $\sup A \leq \inf B$ always holds.


*$f$ is Darboux integrable if and only if $\sup A = \inf B$. (This is often taken as the definition of integrability.)


*By the property of supremum/infimum, there exist sequences $\{ P_n \}$ and $\{ Q_n \}$ in $\mathcal{P}$ such that
$$\lim_{n\to\infty} L(f, P_n) = \sup A \qquad\text{and}\qquad \lim_{n\to\infty} U(f, Q_n) = \inf B. $$
For instance, for each integer $n \geq 1$, pick $P_n, Q_n \in \mathcal{P}$ such that $L(f, P_n) \geq \sup A - \frac{1}{n}$ and $U(f, Q_n) \leq \inf B + \frac{1}{n}$. Also, note that we do not impose any specific conditions on $\{P_n\}$ and $\{Q_n\}$ here.
The definition is then based on the observation that $\{ P_n \}$ and $\{ Q_n \}$ may be chosen so as to satisfy $P_n = Q_n$ for all $n \geq 1$.
To conclude, we mention that the followings are equivalent:

*

*$\sup_{P \in \mathcal{P}} L(f,P) = \inf_{P \in \mathcal{P}} U(f, P)$.


*There exist partitions $\{P_n\}$ and $\{Q_n\}$ such that $U(f, Q_n) - L(f, P_n) \to 0$.


*There exist partitions $\{P_n\}$ such that $U(f, P_n) - L(f, P_n) \to 0$.


*There exist partitions $\{P_n\}$ such that $P_n \subseteq P_{n+1}$ for all $n$ and $U(f, P_n) - L(f, P_n) \to 0$.


*There exist partitions $\{P_n\}$ such that $\| P_n \| \to 0$ and $U(f, P_n) - L(f, P_n) \to 0$.
Here, $\| \{ a = x_0 < x_1 < \dots < x_{n-1} < x_n = b\} \| := \max_{1\leq i \leq n} |x_i - x_{i-1}|$ denotes the mesh size of a partition.
