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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*.