Upper and lower sum - potential error in calculus textbook

Question

In my calculus book the author writes: $$ U(cf, P) =cL(f, P) \quad (1) $$ whereby $\mathbb{R}\ni c < 0$ and $U(cf, P) = \displaystyle\sum_{k=1}^{n} \sup\left\{cf(x)\mid x \in [x_{k-1}, x_k]\right\} \Delta x$ denotes the upper sum, similarly $L(f, P) = \displaystyle\sum_{k=1}^{n} \inf\left\{cf(x)\mid x \in [x_{k-1}, x_k]\right\} \Delta x$ denotes the lower sum.

Shouldn't $(1)$ rather be $$ U(cf, P) =|c|L(f, P) ? $$

Answer 1

Since $c < 0$, we have $c = -|c|$ and

$$\begin{align}\sup\left\{cf(x)\mid x \in [x_{k-1}, x_k]\right\}&= \sup\left\{-|c|f(x)\mid x \in [x_{k-1}, x_k]\right\} \\&\underbrace{=}_{\text{by Property A}} -\inf\left\{|c|f(x)\mid x \in [x_{k-1}, x_k]\right\}\\ &\underbrace{=}_{\text{by Property B}} -|c|\inf\left\{|f(x)\mid x \in [x_{k-1}, x_k]\right\} \end{align}$$

Hence,

$$\begin{align} U(cf,P) &= \sum_{k=1}^n\sup\left\{cf(x)\mid x \in [x_{k-1}, x_k]\right\} \Delta x_k \\&= -\sum_{k=1}^n|c| \inf\left\{f(x)\mid x \in [x_{k-1}, x_k]\right\} \Delta x_k \\ &= -|c| L(f,P) = cL(f,P)\end{align}$$

Here we have used the following properties of supremum and infimum.

Property A: $\sup(S) = -\inf(-S)$ where $-S = \{ a \mid -a \in S\}$

Property B: If $\alpha > 0$, then $\inf(\alpha S) = \alpha \inf(S) $ where $\alpha S = \{\alpha a \mid a \in S\}$.

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Property A is cited in most real analysis texts.

For a proof of Property B, note that for all $a \in S$, $\alpha \inf(S) \leqslant \alpha a $ which implies that $\alpha \inf(S) \leqslant \inf(\alpha S)$. We also have $\inf(\alpha S) \leqslant \alpha a$ which implies that $\alpha^{-1} \inf(\alpha S) \leqslant a \leqslant \inf(S)$ and, thus, $ \inf(\alpha S) \leqslant \alpha \inf(S)$.

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Answer 2

You might have been confusing some point here, so let me first show you that $\sup cA = c\inf A$ for $c < 0$ with $A$ being a set which is bounded above:

Let $x\in A$ be arbitrary an suppose $s:= \sup cA, \ t:= c\inf A$ and since $c < 0$ we find $$ cx \leq s \Longleftrightarrow x \geq \dfrac{s}{c} $$

so we find that $\dfrac{s}{c}$ is a lower bound of $A$ meaning $\dfrac{s}{c}\leq t\Longleftrightarrow ct \leq s\ \ (1)$. Similarily, $$ t \leq x\Longleftrightarrow c t \geq c x $$ meaning $ct$ is an upper bound of $cA$ so $ct\geq s\ \ (2)$. From $(1)$ and $(2)$ the equality follows, so $ct = s \iff c\inf A = \sup cA$.

We can use the fact we have just established to show the property you questioned. $$ \begin{align} &U(cf, P) = U(cf, P) = \sum_{k=1}^{n} \sup\left\{cf(x)\mid x\in [x_{k-1}, x_k]\right\}\Delta x\\ &\stackrel{\text{Property we've just established}}{=} \ \sum_{k=1}^{n} c\inf\left\{f(x)\mid x\in [x_{k-1}, x_k]\right\}\Delta x\\ &= \ c\sum_{k=1}^{n} \inf\left\{f(x)\mid x\in [x_{k-1}, x_k]\right\}\Delta x = cL(f, P). \end{align} $$