Real Analysis/Section 1 Exercises/Hints

These are a list of problems for The Real Numbers section of the wikibook. Most of these problems can be described as algebraic problems, although this problem set also includes theorems and concepts from Number Theory. Number Theory is not a primary topic in this wikibook, but there is an appendix section devoted to formalizing several concepts from it. It is recommended to do some problems from the Number Theory heading since its scope of discussion—theorems related to the natural numbers and its supersets integers and rationals—are seldom discussed clearly and usually left to intuition.

Unsorted

1. Show that $-1<0$
2. Show that $\mathrm{\forall }z\in \mathbb{Z},z^2\ge 0$
3. Complete the proofs of the simple results given above.
4. Show that the complex numbers $ℂ$ cannot be made into an ordered field.
5. Complete the proof of the square roots theorem by giving details for the case $x<1$.
6. Suppose A is a non-empty set of real numbers that is bounded above and let s = sup A. Show that if s is not in A, then for any ε > 0, there exists an element a in A such that s − ε < a < s.

Algebra

The following questions are meant to formalize a good chunk of algebraic rules one may have simply memorized in elementary mathematics as axiomatically true. However, provided with even the first few laws established in the section The Real Numbers such as the commutative law and algebraic manipulations like moving variables around the equal sign, the following questions should be an easy way to get used to applying theorems to prove one's claim—a very important skill in mathematics.

1. Prove the following theorems on inequalities (Assume that the variables, unless explicitly restricted, can be any number in its assumed domain)

1. If 0 ≤ x, then -x ≤ 0
2. If a < b, then -b < -a
3. Given x < 0, if y < z, then xy > xz
4. If a < b and c < d, then a + c < b + d
5. If a < b and c > d, then a - c < b - d
6. If 0 ≤ a < b and 0 ≤ c < d, then ac < bd

2. Prove the following inequalities (Assume that the variables, unless explicitly restricted, can be any number in its assumed domain)

1. If 1 ≤ x, then x ≤ x²
2. If 1 ≤ x, then 1 ≤ x²
3. If 0 < x < 1, then x² < x
4. If 0 ≤ x < y, then x² < y²
5. Given x, y such that 0 ≤ x, y, if x² < y², then x < y
6. Given an odd number n, if x < y, then xⁿ < yⁿ
7. Given a natural number n, if 0 ≤ x < y, then xⁿ < yⁿ

3. Prove the following consequential theorems related to the laws provided in this chapter

1. If there exists the number 0, then $0=-0$
2. $\mathrm{\forall }x,y\in \mathbb{Z},-(xy)=(-x)y=x(-y)$

4. Prove the following theorems on rational numbers

1. Given $a\ne 0,b\ne 0$, $\frac{1}{ab}=\frac{1}{a}\cdot \frac{1}{b}$
2. Given $b\ne 0,c\ne 0$, $\frac{a}{b}=\frac{ac}{bc}$
3. Given $b\ne 0,d\ne 0$, $\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$
4. Given $b\ne 0,d\ne 0$, $\frac{a}{b}÷\frac{c}{d}=\frac{ad}{bc}$

Answers

Absolutes

1. Prove the following inequalities (Assume that the variables, unless restricted, can be any number)

1. |a| + |b| ≤ |a + b|

Answers

Number Theory

1. Prove the following properties on even and odd numbers:
   1. If you add two even numbers, then the sum is even.
   2. If you add two odd numbers, then the sum is even.
   3. If you multiply an odd number with an even number, then the product is even.
   4. If you multiply two odd numbers, then the product is odd.
2. Prove that no consecutive number of a perfect square is also a perfect square for all natural numbers. You don't have to factor in 0 for this problem.
3. Prove that there exists no primitive Pythagorean triple $a,b,c$ such that either a and b are even or a and b are odd.
4. Given that $\mathrm{\forall }a\in \mathbb{Z}:[\mathrm{\exists }b\in \mathbb{N}:\mathrm{\exists }r,q\in \mathbb{Z}:a=bq+r]$, prove that if the given holds, then the remainder r has the following property $0\le r<b$.
5. Prove that $\sqrt{2}$ is irrational.
6. Prove that any square root of a prime number is irrational.
7. Given the equation $x-c=A(x-d)+B(x-e)$ such that $A,B,c,d,e$ are constants, prove that if $x$ is any number except for $d$ or $e$ then both $A$ and $B$ cannot be defined.

Bridge Questions

The following questions can be solved more easily and quickly with more advanced tools. However, solving these questions with restrictions in your mathematical tools provides an excellent understanding of how mathematics as a whole interacts. As a general rule, the answers to these problems should be longer and rely on a lot more properties.

Hints / Answers

1. No Hint
2. No Hint
3. No Hint
4. No Hint
5. No Hint
6. No Hint
7. No Hint
8. The only fact you need use about $ℂ$ is that it contains a square root of $-1$.
9. In the general case you will probably want to divide by your prospective square root, as in the part of the proof which was given, so you might want to treat the case $x=0$ separately.
10. No Hint