# Real Analysis/Section 1 Exercises/Answers

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### Exercises[edit]

- Show that
- Show that
- Show that
- Show that
- Show that
- Let be any prime. Show that is irrational.
- Complete the proofs of the simple results given above.
- Show that the complex numbers cannot be made into an ordered field.
- Complete the proof of the square roots theorem by giving details for the case .
- 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*.

### Hints / **Answers**[edit]

*Answer me**Answer me**Answer me**Answer me**Answer me*- With the view of getting a contradiction, assume is rational. Then for some integers and such that they have no common factor other than one, (that is, and are in lowest terms). Squaring both sides and rearranging terms gives Since is prime, must be divisible by , say for some integer By substitution, , so that , and thus must also be divisible by , a contradiction.
*Answer me**Answer me**Answer me**Answer me*