Textbook confusion regarding "supremum" and "infimum" and "lower bound" and "upper bound"

Question

My textbook, An Introduction to Laplace Transforms and Fourier Series, Second Edition, by Phil Dyke, says the following:

Let $F(x)$ be a function which is defined and is bounded in the interval $a \le x \le b$ and suppose that $m$ and $M$ are respectively the lower and upper bounds of $F(x)$ in this interval (written $[a, b]$ see Appendix C). Take a set of points

$$x_0 = a, x_1, x_2, \dots, x_{r - 1}, x_r, \dots, x_n = b$$

and write $\delta_r = x_r - x_{r - 1}$. Let $M_r, m_r$ be the bounds of $F(x)$ in the subinterval $(x_{r - 1}, x_r)$ and form the sums

$$S = \sum_{r = 1}^n M_r \delta_r$$

$$s = \sum_{r = 1}^n m_r \delta_r$$

These are called respectively the upper and lower Riemann sums corresponding to the mode of subdivision. It is certainly clear that $S \ge s$. There are a variety of ways that can be used to partition the interval $(a, b)$ and each way will have (in general) different $M_r$ and $m_r$ leading to different $S$ and $s$. Let $M$ be the minimum of all possible $M_r$ and $m$ be the maximum of all possible $m_r$. A lower bound or supremum for the set $S$ is therefore $M(b - a)$ and an upper bound or infimum for the set $s$ is $m(b - a)$.

Shouldn’t the sentence

Let $M$ be the minimum of all possible $M_r$ and $m$ be the maximum of all possible $m_r$.

be “$M$ is the maximum of all possible $M_r$ and $m$ is the minimum of all possible $m_r$.”?

And shouldn’t the sentence

A lower bound or supremum for the set $S$ is therefore $M(b - a)$ and an upper bound or infimum for the set $s$ is $m(b - a)$.

be “An upper bound for the set $S$ is therefore $M(b - a)$ and a lower bound for the set $S$ is $m(b - a)$.”?

Using the definitions of of supremum and infimum from Mathematical Analysis by Rudin (see below), the supremum is the least upper bound and the infimum is the greatest lower bound. So not only are supremum and "lower bound" actually different concepts — it seems that they would be contradictory concepts? After all, something cannot be both a supremum and a lower bound. And analogously for infimum and "upper bound”?

I would greatly appreciate it if people would please take the time to clarify this.

Answer 1

I don't have a copy of the textbook, so I can't be completely sure what they're trying to state in that section. However, a key thing about using the upper & lower Riemann sums is dealing with their convergence, if any, to a specific value. In that regard, you want to consider how the smallest values of the upper sums approaches the largest values of the lower sums. As such, the sentence

Let $M$ be the minimum of all possible $M_r$ and $m$ be the maximum of all possible $m_r$.

has an appropriate approach, but as stated in various places, including in the comment to this answer, it's poorly written, but what you're suggesting is not appropriate.

However, you're correct there's a mistake in the next sentence of

A lower bound or supremum for the set $S$ is therefore $M(b - a)$ and an upper bound or infimum for the set $s$ is $m(b - a)$

The terms "supremum" and "infimum" should be switched around in that sentence.

Answer 2

I agree that the explanation, as you have transcribed it, is confusing.

Let $T\subset S$ be a bounded subset. Let the set $$U = \{ M\in S|\ \forall x \in T: x\leq M \}$$ be the set of all the upper bounds $M$ of $T$. We then define the supremum $$\sup T := M'\in U:\forall M\in U,M\geq M' $$

Analogously we take the set of all lower bounds $$L = \{ m\in S|\ \forall x \in T: x\geq m \}$$ and define accordingly the infimum $$\inf T := m'\in L:\forall m\in L,m\leq m' $$

Given this, I'd recommend reading this to understand correctly the construction in your book: https://en.wikipedia.org/wiki/Riemann_integral#Definition