• Oct 31st 2012, 12:08 PM
AU11
Hi,

I have to prove the following by contradiction

the set S = [all natural numbers n such that n is a multiple of 13] has no greatest element

I have started by saying let greatest value = x, where x is a member of S and x is greater than or equal to n

then let n=13k for some natural number k

x is greater than or equal to 13k

not sure where to go from here??

thanks
• Oct 31st 2012, 12:48 PM
Plato
Quote:

Originally Posted by AU11
I have to prove the following by contradiction
the set S = [all natural numbers n such that n is a multiple of 13] has no greatest element

We know that $\displaystyle (\forall k\in\mathbb{N})[k<k+1]~.$

If $\displaystyle n=\max(S)$ is it true $\displaystyle n=13k~\&~n=13k<13(k+1)=n+13~?$
• Oct 31st 2012, 12:54 PM
AU11
Hi,

I don't get why you have written k=13k?

plus do I not talk about x (the greatest element) anymore?
• Oct 31st 2012, 02:02 PM
Plato
Quote:

Originally Posted by AU11
Hi,

I don't get why you have written k=13k?

plus do I not talk about x (the greatest element) anymore?

See my edit.

If $\displaystyle x$ is a real number $\displaystyle x+1$ is a real number and $\displaystyle x<x+1$ so there cannot a greatest real number.

If $\displaystyle 0<x$ then $\displaystyle 0<\frac{x}{2}<x$ so there can be no smallest positive number.
• Nov 2nd 2012, 07:12 AM
AU11