TITLE: The equivalence of definitions Riemann integral
QUESTION [9 upvotes]: First definition of Riemann integrable function. Let $f:[a,b] \to \mathbb{R}$ be a bounded function and $P=\{x_0,x_1,\dots, x_n\}$ partition of $[a,b]$. Define $U(P,f):=\sum \limits_{i=1}^{n}M_i\Delta x_i$ and $L(P,f):=\sum \limits_{i=1}^{n}m_i\Delta x_i$ where $M_i=\sup\limits_{[x_{i-1},x_i]} f(x),\quad m_i=\inf\limits_{[x_{i-1},x_i]} f(x), \Delta x_i=x_i-x_{i-1}.$ Let $\inf \limits_{P}U(P,f)=I^*$ and $\sup \limits_{P}L(P,f)=I_*$. If $I^*=I_*=I$ then we called $f(x)$ is Riemann integrable function on $[a,b]$ with integral $I$.
Second definition of Riemann integrable function. Let $P=\{x_0,x_1,\dots, x_n\}$ is a partition of $[a,b]$ with $\xi_i\in [x_{i-1},x_i]$ and we define Riemann-integral sum $\sigma(P):=\sum \limits_{i=1}^{n}f(\xi_i)\Delta x_i$ and $\lVert P\rVert=\max\limits_{i}\Delta x_i$. If the following limit $\lim \limits_{\lVert P\rVert\to 0}\sigma(P)$ exists and has value $L$ we say that $f(x)$ is Riemann integrable function on $[a,b]$ with integral $L$.
The first definition is from Rudin's PMA book but in other books I met the second definition. But I can't prove the equivalence of these definitions for a couple days. Can anyone show to me a strict and rigorous proof? I would be very grateful for your help!
P.S. Happy New Year! :)

REPLY [5 votes]: This is explained very nicely in Apostol's Mathematical Analysis. In this book Apostol uses Riemann sums to define the definite integral, but he does not use the limit based on $||P|| \to 0$. Rather he uses the concept of finer partition.
Thus if $P, P'$ are partitions of $[a, b]$ then $P'$ is said to be finer than $P$ if $P \subseteq P'$. Adding more points to an existing partition makes it finer. The other concept is called norm of a partition which is defined as the length of largest sub-interval made by the partition. When you add points to a partition to make it finer, then the norm can only decrease (or remain unchanged). Thus finer partitions corresponds to partitions with smaller norms, but the converse does not hold.
It appears from Apostol's presentation that dealing with a limit where norm of partition tends to $0$ is difficult compared to dealing with limit when partitions become finer and finer. And he uses the following definition of Riemann integral:
Let $f$ be bounded on $[a, b]$ and let $P = \{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\}$ be a partition of $[a, b]$. A sum of the form $$S(P, f) = \sum_{i = 1}^{n}f(\xi_{i})(x_{i} - x_{i - 1})$$ where $\xi_{i}$ is any point in the interval $[x_{i - 1}, x_{i}]$ is called a Riemann sum of $f$ over partition $P$. A number $I$ said to be the Riemann integral of $f$ over $[a, b]$ and we write $$\int_{a}^{b}f(x)\,dx = I$$ if for every $\epsilon > 0$ there is a partition $P_{\epsilon}$ of $[a, b]$ such that $$|S(P, f) - I| < \epsilon$$ for partitions $P$ of $[a, b]$ which are finer than $P_{\epsilon}$.
It is now easy to show the equivalence of this definition of Riemann integral with the definition based on Darboux sums (the first definition of your question). This is because the Riemann sums are always sandwiched between the upper and lower Darboux sums and as partitions get finer and finer the lower sums increase and upper sums decrease. If these Darboux sums have same limit (i.e. supremum of lower sums is equal to infimum of upper sums) then obviously the Riemann sums also tend to same limit as partitions become finer and finer. On the other hand if $M_{i}, m_{i}$ are the supremum and infimum of $f$ on a sub-interval $[x_{i - 1}, x_{i}]$ generated by partition then it is easy to choose points $\xi_{i}, \xi'_{i}$ in this sub-interval such that $f(\xi_{i})$ is near $M_{i}$ and $f(\xi'_{i})$ is near $m_{i}$. Due to this we can a find a Riemann sum close to $U(P, f)$ and another Riemann sum close to $L(P, f)$. So if Riemann sums tends to a limit when partitions get finer and finer then the upper and lower Darboux sums also tend to the same limit.
Next Apostol shows the equivalence of his definition (mentioned above) with the definition based on limit of type $||P|| \to 0$ in his exercise. This proof is also available as an answer here.
Also note that in general (i.e. in the context of Riemann-Stieltjes integral) these two definitions are not equivalent and the one which uses the concept of finer partitions is more inclusive than the one based on norm. See this answer for more details.