**1. The problem statement, all variables and given/known data**
Given x<y for some real numbers x and y. Prove that there is at least

one real z satisfying x<z<y

**2. Relevant equations**
This is an exercise from Apostol's Calculus Vol. 1. The usual laws of

algebra, inequalities, a brief discussion on supremum, infimum and

the Archimedean property preceeded this exercise.

**3. The attempt at a solution**
I'm trying to solve as many problems from the first chapter, which I had

just glossed over during a first read. Proving theorems in a rigorous

fashion is not exactly my forte. It becomes all the more difficult when

I'm asked to prove something "intuitively obvious."

I think I have the solution in fragments.

Let S be the set of all numbers greater than x. x is a lower bound for

this set, and therefore S has a greatest lower bound - inf S.

[tex]inf S \geq x [/tex]

[The following argument assumes that x = inf S]

Consider some [tex]y \in S[/tex]. I can always find at least one [tex]z \in S[/tex] such that z < y, because if there are no z's satisfying that inequality then y would be inf S.

The only problem here seems to be to prove that inf S = x. The first choice is a proof by contradiction.

Assume inf S > x...

Am I on the right track?

**1. The problem statement, all variables and given/known data**
**2. Relevant equations**
**3. The attempt at a solution**
**1. The problem statement, all variables and given/known data**
**2. Relevant equations**
**3. The attempt at a solution**
**1. The problem statement, all variables and given/known data**
**2. Relevant equations**
**3. The attempt at a solution**
**1. The problem statement, all variables and given/known data**
**2. Relevant equations**
**3. The attempt at a solution**