I am self-learning Real Analysis from the text Understanding Analysis, by Stephen Abbott. I am trying to prove that two characterizations of the Riemann integral are equivalent. The stated definition (7.2.7) in the book for the Riemann Integrability is:
Definition. (Riemann Integrability). A bounded function $f$ defined on the interval $[a,b]$ is a Riemann-integrable if the upper and lower integrals are equal i.e.
$$\sup \{ L(f,P): P \in \mathcal{P} \} = L(f) = U(f) = \inf \{ U(f,P): P \in \mathcal{P}\}$$
Exercise problem 7.2.6 asks to prove that if the Riemann sums of $f$, with respect to a tagged partition converge, then so do the lower and upper Riemann sums.
I thought over a few days about this question, but I can't seem to make any meaningful progress. I wrote down my initial thoughts. Do you have a hint/clue on how to proceed, without revealing the proof/giving away the solution?
[Abbott 7.2.6] A \textit{tagged} partition $\displaystyle ( P,\{c_{k}\})$ is one where in addition to a partition $\displaystyle P$ we choose a sampling point $\displaystyle c_{k}$ in each of the subintervals $\displaystyle [ x_{k-1} ,x_{k}]$. The corresponding Riemann sum:
\begin{align*} R(f,P) =\sum_{k=1}^{n} f( c_{k}) \Delta x_{k} \end{align*}
is discussed in section 7.1., where the original definition is alluded to.
Riemann's original definition of the Integral. A bounded function $f$ is integrable on $\displaystyle [a,b]$ with $\int_{a}^{b} f=A$ if for all $\displaystyle \epsilon >0$ there exists a $\displaystyle \delta >0$ such that for any tagged partition $\displaystyle ( P,\\{c_{k}\\})$ satisfying $\displaystyle \Delta x_{k} < \delta $ for all $\displaystyle k$, it follows that
\begin{align*} |R( f,P) -A|< \epsilon \end{align*}
Show that if $\displaystyle f$ satisfies Riemann's definition above, then $\displaystyle f$ is integrable in the sense of the definition 7.2.7. (The full equivalence of these two characterizations of integrability is proved in section 8.1).
Initial thoughts.
Clearly, since $m_k \leq f(c_k) \leq M_k$, for any arbitrary partition $P$, it follows that:
\begin{align*} L(f,P) \leq R(f,P) \leq U(f,P) \end{align*}
I need to show that distance $U(f,P) - L(f,P)$ can be made arbitrarily small.
Preliminaries:
From Riemann's definition, for any $\epsilon > 0$ there exists a partition $P=(x_0,x_1,\ldots,x_n)$ of $[a,b]$ such that for any choice of tags, the Riemann sum $R(f,P)$ satisfies $$\tag{*}A - \frac{\epsilon}{4} < R(f,P) < A + \frac{\epsilon}{4}$$
The upper and lower sums are defined as
$$U(f,P) = \sum_{j=1}^n M_j(x_j-x_{j-1}), \quad L(f,P) = \sum_{j=1}^n m_j(x_j-x_{j-1})$$
where $M_j = \sup\{f(x): x \in [x_{j-1},x_j]\}$ and $m_j = \inf\{f(x): x \in [x_{j-1},x_j]\}$.
Hint:
By properties of the supremum and infimum, there exist $\xi_j, \eta_j \in [x_{j-1},x_j]$ such that
$$M_j - \frac{\epsilon}{4(b-a)} < f(\xi_j), \quad f(\eta_j) < m_j + \frac{\epsilon}{4(b-a)} $$
Use this to relate upper and lower sums to tagged Riemann sums for partition $P$ and then show $U(f,P) - L(f,P)< \ldots$