Thread: Proving inequality For every integer n > 0 there is some real number x > 0 s.t. x < 1

For every integer n > 0 there is some real number x > 0 s.t. x < 1/n..

I need a proof. Contrapositive? I don't know what to do.

Thanks.





Another question, too:

A fraternity has a rule for new members: each must always tell the truth or always lie. They know
which does which. If I meet three of them on the street and they make the statements below,
which ones (if any) should I believe?
A says: \All three of us are liars."
B says: \Exactly two of us are liars."
C says: \The other two are liars."

Maybe you want to say an irrational number because is a real number such that

Let be an irrational number. Then the sequence decreases and has for limit. So there is a such that
We just have to prove that is irrational: if that wasn't the case, since then would be a rational number, contradiction. Thus is an irrational number, and since there exist positive irrational numbers, that's ok.

An example:


At first sight, I would believe B

If A tells the truth, then he lies... Impossible
If C tells the truth (which implies that B lies), then B is right... Impossible
If B lies, then A tells the truth... Impossible