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neutrino
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Homework Statement
Given x<y for some real numbers x and y. Prove that there is at least
one real z satisfying x<z<y
Homework 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.
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?