Real Analysis/Section 1 Exercises/Hints

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Real Analysis/Section 1 Exercises
Exercises

General[edit]

  1. Show that -1<0\
  2. Show that \forall x,y\in\mathbb{Z},\ -(xy)=(-x)y=x(-y)
  3. Show that \forall z\in\mathbb{Z},\ z^2\ge0
  4. Show that \forall z\in\mathbb{Z},\ z\ge0\Leftrightarrow-z\le0
  5. Show that \forall x,y,z\in\mathbb{Z},\ (x<0)\cap(y<z)\Leftrightarrow(xy>xz)
  6. Let p be any prime. Show that \sqrt{p} is irrational.
  7. Complete the proofs of the simple results given above.
  8. Show that the complex numbers \mathbb C cannot be made into an ordered field.
  9. Complete the proof of the square roots theorem by giving details for the case x<1.
  10. 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.

Inequalities[edit]

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

  1. If a < b, then -b < -a
  2. If a < b and c < d, then a + c < b + d
  3. If a < b and c > d, then a - c < b - d
  4. If 1 ≤ x, then x ≤ x2
  5. If 1 ≤ x, then 1 ≤ x2
  6. If 0 < x < 1, then x2 < x
  7. If 0 ≤ a < b and 0 ≤ c < d, then ac < bd
  8. If 0 ≤ x < y, then x2 < y2
  9. Given x, y such that 0 ≤ x, y, if x2 < y2, then x < y
  10. Given an odd number n, if x < y, then xn < yn
  11. Given a natural number n, if 0 ≤ x < y, then xn < yn
Methodology Hint (General 1)

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

Further Reading (I)

This is the proof of why the inequality sign "flips" when you multiply by -1, a valid interpretation for what you're doing.

Answers

Absolutes[edit]

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 (I)

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

Bridge Questions[edit]

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[edit]

  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