1. The problem statement, all variables and given/known data
We just started learning about supremums and infimums in my math proofs class. I am having trouble with the following question:
Let x, y be real numbers with y - x > 1. Prove that there exists an integer n such that x < n < y. Hint--use the well ordering principle.
2. Relevant equations
Well-ordering principle: a nonempty set S of real numbers is said to be well-ordered if every nonempty subset of S has a least element. This is proved using induction.
Upper bound: a real number m is called an upper bound of S if for every s in S, s <= m.
Lower bound: a real number m is called an lower bound of S if for every s in S, m <= s.
Maximum: If m is an upper bound of S and m is in S, then we call m the maximum of S.
Minimum: If m is a lower bound of S and m is in S, then we call m the minimum of S.
Supremum: least upper bound
Infimum: greatest lower bound
3. The attempt at a solution
I am unsure of how to incorporate the well ordering principle into my proof. I was thinking of writing something like, "y - x > 1 <==> y > x + 1 <==> x < x + 1 < y. So there is an integer z such that x <= z < x + 1 or x < z <= x + 1." And then show the rest with ceiling and floor... but I'm sure this is incorrect, as we are supposed to use the well ordering principle.
Please let me know if extra clarification is needed.
Any help would be GREATLY appreciated!
Thanks in advance,