Need help with analysis questions...

Nappy

1: Give a complete, formal proof of the following statement: If a E R, then there exists a negative

integer n such that n < a.

where R= real numbers and E = is an element

I found most of these analysis questions extremely hard as I dont know where to start with them?? Can anyone give me a helping hand to make a start on this one...

Cheers

Nappy

Thomas_S_Hunterson

This page may help:
http://en.wikipedia.org/wiki/Floor_and_ceiling_functions

/edit: It's not identical, kind of related, but possibly confusing things.

Nappy

Well as the real numbers is just the number line from negative infinity to positive infinity the statement seems obviously correct, But how to prove it?? All I can really take from above link is that n< floor a but dont know how to transform all this into a formal proof.

Thanks for help

Fremen

Presumably you were given a list of axioms for the real numbers in the context of this class. You need to use combinations of these axioms along with proof by contradiction to derive the result.
If you post up the axioms I can help you more, of you're still stuck.

ZorbaTehZ

If its [latex]n \in Q[/latex] then you can draw parallels from the theorem about integers that says given any [latex]x \in R[/latex] there is always some [latex]n \in N[/latex] such that [latex]n > x[/latex]. If you start assuming the converse is true, ie. [latex]n \geq a\ \forall_{n \in Q}[/latex] then you should be easily able to come up a contradiction - also you don't need that floor/ceiling stuff above at all to prove it.