Long(er)-Form Mathematics

Real Analysis: Hints and Solutions

For part (b), prove this by finding a counterexample in the case that $\mathbb{F} = \mathbb{R}$.

For part (b), prove this by finding a counterexample in the case that max$\{x,y,z\} = \text{max}\{x,\text{max}\{y,z\}\}$.

Use induction with the triangle inequality.

When you are trying to prove something holds for every positive integer, that is a good indication that induction is worth trying.

First prove that if $A$ is a subset of $\mathbb{N}$ that is bounded above, then $A$ must be finite. Then apply Exercise 1.25. Then apply Exercise 1.24.

You might find it easiest to prove (c) $\Rightarrow$ (b) $\Rightarrow$ (a) $\Rightarrow$ (c).

For part (b), note that every interval contains a rational number.

See if you can find an uncountable collection of “simple” functions. If you can do that, there’s no need to even consider complicated functions like $e^x$ and $\sin(x)$. See if you can start with a set that you know to be uncountable, and create a distinct function for each element in that set.

Begin by writing a set as a 0-1 sequence where the $i^{\text{th}}$ number is 0 if $i$ is in the set, and 1 if it’s not in the set.

Each point on the unit circle can be identified by an angle.

For $0<r<1$ , show that the sequences is monotone decreasing and bounded below; then apply a theorem. Then using what you proved, prove the case $-1<r<0$. For $r>1$ show that the sequence is monotone increasing and unbounded; then apply a theorem. For $r\leq -1$ , use what you proved as well as the fact that when you multiply a number by a negative, it switches signs.

First think about the case where $(y_n)$ and $(z_n)$ are both monotone increasing.

For both, consider using the comparison test.

To prove the implication $\Leftarrow$, imitate the proof that the harmonic series diverges.

Consider the partial sums $(s_n)$ of the given series, and the partial sums $(s_n’)$ of a rearrangement of the given series, and show that $|s_n−s_n’| \to 0$ .

Use the observation that $a_k = s_k – s_{k-1}$.

Use DeMorgan’s Laws.

For the reverse direction, consider using the contrapositive.

Consider the complements of the sets in $\mathcal{S}$.

First prove that for every $x\in \mathbb{R}$ , the singleton set ${x}$ is closed. What does this imply?

For a point $a\in A$ , consider the set $\{x\in \mathbb{R} : (a,x) \subseteq A\}$. Prove that if this set has a supremum, then it belongs to the boundary of $A$ .

First show that if $f_1$ and $f_2$ are continuous, then so is max$\{f_1,f_2\}$.

This exercise asks you to prove the existence of two special points, $c$ and $d$. We have two theorems in this chapter whose conclusions are about the existence of a point with a special property (the intermediate value theorem and the extreme value theorem). And as my own undergraduate real analysis professor Jamie Radcliffe used to tell us, “use your theorems!”

In the proof of the intermediate value theorem we used a trick to generate a function $g$ which gave us the result. Try something similar.

First prove it in the case that $g(x) = 0$ for all $x$.

This exercise asks you to prove the existence of a point with a special property. We have two theorems in this chapter whose conclusions are about the existence of a point with a special property (the intermediate value theorem and the extreme value theorem). And as my own undergraduate real analysis professor Jamie Radcliffe used to tell us, “use your theorems!”

If you apply the right theorem, there’s a 1-line proof.

This exercise asks you to prove the existence of a point with a special property. We have two theorems in this chapter whose conclusions are about the existence of a point with a special property (the intermediate value theorem and the extreme value theorem). And as my own undergraduate real analysis professor Jamie Radcliffe used to tell us, “use your theorems!”

If a function $f$ is defined on a domain of, say, $[0,1]$, you can create another function $g$ on a domain of, say, $[0,2]$ which matches $f$ on $[0,1]$ and equals, say, 0 or 1 on $(1,2]$.

This exercise asks you to prove the existence of a point with a special property. We have two theorems in this chapter whose conclusions are about the existence of a point with a special property (the intermediate value theorem and the extreme value theorem). And as my own undergraduate real analysis professor Jamie Radcliffe used to tell us, “use your theorems!”

Use one of the really big theorems from this chapter. (You know a theorem’s big if it was given a legit name.)

For part (a), each discontinuity must results in a jump discontinuity. Notice that when that happens, there is some interval of $y$-values that are jumped over. And every interval contains a rational number. For part (b), consider assuming for a contradiction that $f$ is not strictly monotone increasing. Recall that, because $f$ is continuous, you can apply the intermediate value theorem.

Exercise 7.4

Differentiate.

One proof is similar in style to Theorem 7.11.

Note since ff is a higher degree polynomial than any of its derivatives, $(f+f’+f”+\dots+f^{(n)})$ also has tails diverging to $\infty$ and hence a minimum value. If you can show that this minimum value is at least 0, then the entire function will be too.

Consider the function $h(x) = g(x) – f(x)$.

“Use your theorems!”

It suffices to prove that $g'(c) > 0$ for each $c \in (0,\infty)$. So, first take the derivative of $g$ and see what you need to prove in terms of $f$. And remember your theorems!

This is Corollary 8.15. Remember that corollaries follow quickly from theorems, so you should probably try to use the previous theorem to prove this…

I. First prove by induction that $\displaystyle \sum_{k=1}^{n} k = \frac{n(n+1)}{2}$.
II. First prove by induction that $\displaystyle \sum_{k=1}^{n} k = \frac{n(n+1)(2n+1)}{6}$.
III. First prove by induction that $\displaystyle \sum_{k=1}^{n} k = \frac{n^2(n+1)^2}{4}$.

Reread the proof that if $f$ being continuous implies $f$ is integrable. One proof of this exercise is similar to that.

For part (b): One slick proof uses part (a) and properties of integrals.

Since f is continuous on $[a,b]$ , it is bounded. That is, there exists some $m$ and $M$ such that $m\leq f(x)\leq M$ for all $x\in [a,b]$. Now consider $f(x) \cdot \left(\int_a^b g(x) \text{dx}\right)$ as a function of $x$ , and recall what we assumed about $f$.

Use part (a) for a quick solution to part (b). For part (d), note that by part (c) the function can’t be continuous (everywhere). However, try to come up with a fairly simply function. Just because it is not continuous (everywhere) doesn’t mean it has to get too crazy.

For part (a), the answer is not $xf(x)$. For part (b), try using part (a).

Take the derivative of both sides.

Try to prove that $\displaystyle \int_1^a 1/x \text{dx} = \int_a^{ab} 1/x \text{dx}$. Think about the definition of an upper or lower sum.

Exercise 8.31 is a special case of this, so start by understanding that one.

For part (b): For a fixed $y$, differentiate $x \mapsto \text{L}(xy)$.

Note for any $\delta > 0$, when restricted to the domain $[a, x_o – \delta]$ or to $[x_0+\delta, b]$, that $f$ is a continuous function.

To prove part (f), note that it is equivalent to showing that there does not exist an interval $[s,t] \subseteq [a,b]$ on which there are no points of continuity. And remember, $f$ being integrable on $[a,b]$ implies it is also integrable on any subinterval of $[a,b]$.

Begin with a step function as in Exercise 8.42.

Try to find a sequence of functions each of whose integral is exactly 1. But try to have them converge pointwise to $f(x) = 0$.

For part (b), you can’t plug in $x=1$ because part (a) only works for $x \in (-1,1)$ … But can you plug something else in?

Here’s one answer (without proof): Let $A=[0,1]$ and

\[f_k(x) = \begin{cases}1/k & \text{ if } 1/2^k < x \leq 1/2^{k-1},\\

0 & \text{otherwise.}\end{cases}\]

Use your theorems.

For part (c): Solve for $f(x)$ and compare coefficients.

During Fall 2025 and Spring 2026 I am teaching real analysis. During these semester I will write up homeworks and solutions and link to them below.