Harrison Chapman

Math 317: Homework 10

Due: Friday, April 26, 2019
1. (33.2) Let $S$ be a nonempty bounded subset of $\mathbb R$. For fixed $c > 0$, let $cS = \{ cs : s \in S \}$. Show that $\sup(cS) = c\cdot\sup(S)$ and $\inf(cS) = c\cdot\inf(S)$.

Proof. Let $M = \sup S$. Then $cM$ is an upper bound for $cS$ as for all $cs \in cS$ we have that $M \ge s$ so $cM \ge cs$ as $c > 0$.

Now, if $\alpha \ge cs$ for all $cs \in cS$ (that is, $\alpha$ is an upper bound for $cS$), then $\alpha/c \ge s$ for all $s \in S$ so $\alpha$ is an upper bound for $S$. But then $M \le \alpha/c$ as $M$ is the least upper bound for $S$, so $cM \le \alpha$. So $cM = c\sup S = \sup(cS)$.

Now notice that $\inf(cS) = -\sup(-cS) = -c\sup(-S) = c\inf S$. $\Box$

2. (33.6) Let $f$ be integrable on $[a,b]$. Prove that, for any subset $S \subseteq [a, b]$ we have

Hint. For $x_0, y_0 \in S$, we have $\vert f(x_0)\vert - \vert f(y_0)\vert \le \vert f(x_0) - f(y_0)\vert \le M(f, S) - m(f, S)$.

Proof. The first part of the hint follows from the triangle inequality, as

The second part of the hint comes from how $M(f, S) \ge f(x_0)$ and $m(f, S) \le f(y_0)$ for all $x_0, y_0 \in S$.

On the other hand, we have that for any $\epsilon > 0$, properties of the supremum and infimum guarantee that there exists $x_0, y_0 \in S$ so that,

So we have that,

This holds for all $\epsilon > 0$, so we conclude that the desired inequality holds. $\Box$

3. (33.5) Show that $\left\vert \int_{-2\pi}^{2\pi} x^2 \sin^8(e^x)\; dx \right\vert \le \frac{16\pi^3}{3}$.

Proof. Using Theorems 33.4 and 33.5, together with the FTC I (for example), we have that (as $\left\vert \sin^8(e^x )\right\vert \le 1$)

4. (33.8) Let $f$ and $g$ be integrable functions on $[a, b]$.

1. It is a fact (see Exercise 33.7) that if $h$ is integrable on $[a,b]$, then so is $h^2$. Prove that $fg$ is integrable on $[a, b]$. Hint. Use that $4fg = (f+g)^2 - (f-g)^2$.

2. Show that $\max(f,g)$ and $\min(f,g)$ are integrable on $[a, b]$. You may use the results of Excercise 17.8 without proof.

Use that for integrable functions $f, g$ we have that $f+g$, $f-g$, $(f+g)^2$, $(f-g)^2$, $\vert f-g \vert$ are all integrable. Further applications of linearity of the integral prove (a), and together with $\min(f,g) = \frac 12 (f+g) - \frac 12 \vert f-g \vert$ prove the first part of (b). The second part then follows from $\max(f,g) = -\min(-f,-g)$ and integrability of $-f$, etc.

5. (33.10) Let $f$ be the function,

Prove that $f$ is integrable on $[-1, 1]$. Hint. See Excercise 33.11(c) and its solution in the textbook. (Why can we not apply the Dominated Convergence Theorem to prove that $f$ is integrable?)

Proof. Let $\epsilon > 0$. Notice that $f$ is continuous on each of $[-1, -\epsilon/8]$ and $[\epsilon/8, 1]$, and is hence integrable. This means that there exist partitions $Q_1$ of $[-1, -\epsilon/8]$ and $Q_2$ of $[\epsilon/8, 1]$ so that

Notice that $P = Q_1 \cup Q_2$ is a partition of $[-1, 1]$ and furthermore that

and,

(Why? The only terms in the two sums are those contributed by the interval $[-\epsilon/8, \epsilon/8]$ but we know that the supremum and infimum of $f$ on any such interval are $1$ and $-1$ respectively and that the width of this interval is $\epsilon/4$) So,

We can find any such $P$ for any $\epsilon > 0$, so we conclude that $f$ is integrable on $[-1, 1]$.