- #1
QuarkCharmer
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Homework Statement
(0,100) has a cover that consists of a finite number of closed interval subsets.
I'm really lost with this one. I can clearly understand why the statement is false, but I'm not sure my proof is good.
Homework Equations
The Attempt at a Solution
Clearly this is false, so I am trying to disprove it.
Proof:
Let S =(0,100) and let C be a cover of S.
Since C contains finitely many closed interval subsets of S,
C has a least element.
Let X_i =[a,b][itex]\in[/itex]C be a subset of S, [itex]\forall[/itex]a,b[itex]\in[/itex]R^+
such that 0<a<b<100
Since there is no smallest positive real number, (which i have proved before),
there is an infinite number of X_i's in C.
But C has a finite number of elements.
This is a contradiction.
Q.E.D.