Here are two great items to prove about our number systems (they may seem intuitive at first, but rigorously proving them can be challenging). See if you can answer them.
1) Prove that between any two real numbers there exists a rational number. Similarly, prove that between any two real numbers there exists an irrational number
Hint: Use Archimedean property.
2) Prove that the union of a finite number of countable sets is countable.