I'm dusting off the cobwebs by going through Rudin for the first time in awhile. My question is, how do we recover the definition of the Riemann integral seen in calculus using Riemann's Criterion (Theorem 6.6) and Theorem 6.7 (In particular part 3).
If $ f$ is integrable on $ $ on $[a,b] $, then for all $ \varepsilon>0$ there exists a partition $ P$ such that $ U(P,f)-L(P,f)<\varepsilon$. Theorem 6.7 tells us that for this $P$ we will also have $$ \left\lvert \sum_{i=1}^nf(t_i) \, \Delta x_i-\int_a^b f(x)\ dx\right\rvert<\varepsilon,$$ if we write $P=\{ x_0, \ldots, x_n\} $ and let $ t_i\in[x_{i-1},x_{i}]$. I can see how this means there is some sequence of the form $\{\sum f(t_i)\, \Delta x_i\} $ which converges to the integral, but I'm struggling to formalize it to conclude $$\lim_{n\to\infty}\sum_{i=1}^nf(t_i) \, \Delta x_i=\int_a^b f(x)\ dx.$$ The limiting behavior happens as a result of the refinement of partitions, not because some number $n\to\infty $. Are we allowed to simply let $P=\{x_0,\ldots,x_n\} $ include more and more points (in a sense letting $n\to\infty $), and make this conclusion? My worry is that the Riemann Criterion simply says there is some partition $P$ for a fixed $ \varepsilon$. It never says that the partition for a smaller value of $\varepsilon$ is written as a refinement of $P$.
This is not so difficult if you realize that the Riemann sum, denoted by $S(f, P, T_P)$ ($T_P$ denoting set of tags $t_i$ corresponding to partition $P$), lies between $U(f, P) $ and $L(f, P) $ and the value of integral, say $I=\int_a^b f(x) \, dx$ also lies between these two Darboux sums. And therefore if $P$ is a partition for which $U(f, P) - L(f, P) <\epsilon $ then for the same partition $P$ we have $$|S(f, P, T_p) - I|\leq U(f, P) - L(f, P) <\epsilon \tag{1}$$ This means that for every $\epsilon >0$ we have a corresponding partition $P$ such that inequality $(1)$ holds.
Next we need to see how this condition is related to the definition of Riemann integral as a limit of Riemann sums as 1) partitions get finer and finer and as 2) norm of partition tends to $0$. The definition based on refinement of partitions is easy as making a partition finer guarantees that the difference $U(f, P) - L(f, P) $ either decreases or stays same and hence the inequality $(1)$ can be ensured for all finer partitions.
It is somewhat difficult to show that under the same conditions we have the following:
On the other hand your worry about partitions getting finer and finer for smaller and smaller $\epsilon $ is unfounded. The definition of limit works like "for every something there is a corresponding another thing..." and in general there is no monotone relation between "something" and "another thing". We just need the existence and not a monotone trend.