# Can anyone help me with this proof?

• Aug 11th 2010, 11:44 AM
Nappy
Can anyone help me with this proof?
Give a complete, formal proof of the following statement: If
a is an element of the real numbers, then there exists a negative

integer
n such that n < a.

I know its related to the archimedean property but cant find the proof anywhere.

Cheers!
• Aug 11th 2010, 11:58 AM
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Quote:

Originally Posted by Nappy
Give a complete, formal proof of the following statement: If
a is an element of the real numbers, then there exists a negative

integer
n such that n < a.

I know its related to the archimedean property but cant find the proof anywhere.

Cheers!

Maybe my question is dumb: Why can't we just take $\displaystyle n=\lceil a\rceil-1$?
• Aug 11th 2010, 12:01 PM
Plato
Quote:

Originally Posted by Nappy
Give a complete, formal proof of the following statement: If a is an element of the real numbers, then there exists a negative integer n such that n < a. I know its related to the archimedean property but cant find the proof anywhere.

Clearly if $\displaystyle a\ge 0$ then $\displaystyle -1<a$ works.
So suppose that $\displaystyle a<0$ then $\displaystyle 0<-a$.
Using the Archimedean property, we know $\displaystyle \left( {\exists N \in \mathbb{Z}^ + } \right)\left[ { - a < N} \right]$.
You finish.