Real Analysis/Section 1 Exercises/Hints

← Properties of The Real Numbers

Real Analysis/Section 1 Exercises Exercises

Sequences→

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 $\forall z\in \mathbb {Z} ,\ z^{2}\geq 0$
3. Complete the proofs of the simple results given above.
4. Show that the complex numbers $\mathbb {C}$ 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. $\forall x,y\in \mathbb {Z} ,\ -(xy)=(-x)y=x(-y)$

4. Prove the following theorems on rational numbers

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

Methodology Hint (General 1)

Most of these shouldn't be too challenging; remember your algebra laws! They are still valid axioms, even in inequalities.

Methodology Hint (General 2a)

If conciliation, these questions are here to reinforce one special property that you can do when solving an inequality problem.

Methodology Hint (General 2b)

As a continuation of Hint 2a, this property is similar to how you can sub in variables with equations. Yet, it still works with inequalities, provided one change.

Hint (Question III)

You can use the alternate definition of x as proven in Question II. Also, remember that you don't need to do everything in one step.

Comments and Further Reading

The question asked in 1ii offers a way to prove why the inequality sign "flips" when you multiply by -1.

The question asked in 1vii is very important for inequality problems where inserting inequalities in between other inequalities is usually not a valid operation. It provides an example of a situation where it is valid.

Answers

Absolutes

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

1. |a| + |b| ≤ |a + b|

Methodology Hint (General 1)

Remember your algebra laws! They are still valid axioms, even in inequalities.

Methodology Hint (General 2a)

Absolute values are positive, by definition.

Methodology Hint (General 2b)

There is one algebraic operation that can guarantee a positive output and whose inverse on the output number is best expressed as an absolute value (in elementary mathematics, it is broken down into cases, if logical, instead).

Further Reading (1i)

This is the proof of the Triangle Inequality. Note that this version applies to the real number line, but the general version shown on that webpage is a generalization which also works.

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 $\forall a\in \mathbb {Z} :[\exists b\in \mathbb {N} :\exists r,q\in \mathbb {Z} :a=bq+r]$, prove that if the given holds, then the remainder r has the following property $0\leq r<b$.
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.

Hint (Question 2 pt.1)

Try writing out what you are trying to prove in math notation instead of tackling it, in its entirety, in your head.

Hint (Question 2 pt.2)

A consecutive number is defined, in laymen terms, for natural numbers as the number next to the number in question using the natural number's total ordering property.

Hint (Question 6 pt.1)

What is the definition of an irrational number? Not rational, which implies that if we assume that it is a rational number then something should contradict.

Hint (Question 6 pt.2)

A prime number is any natural number greater than 1 that only has the possible prime factor of itself, which can be alternatively expressed as $p=1*p$ such that p is the prime number. 1 is used to highlight the factor p, but 1 is not technically a factor.

Hint (Question 7)

Try defining the constant A or B and take note of what each variable represents.

Further Reading (5)

The proof that the square root of 2 is irrational is a famous proof that is relatively easy to solve compared to other numbers, as this can be easily proved using a coprime rational number—a usually assumed property of a rational number to begin with.

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.

1. Given a natural number n, prove that $|x|<x^{n}$

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 $\mathbb {C}$ 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