Thread: Can anyone help me with this proof?

1. 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!

2. 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 $\displaystyle n=\lceil a\rceil-1$?

3. 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.