# An ordered field has no smallest positive element

Prove that every ordered field has no smallest positive element.

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@Ross: I just removed it. –  Brian M. Scott May 8 '12 at 3:04

Hint: Suppose that $x$ is the smallest positive element. Show that $1/2$ is positive, and thus $x/2$ is positive. Thus $0<x/2<x$, a contradiction.

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A few questions about this. What if the field is finite or has finite characteristic? –  nullUser May 8 '12 at 3:10
A field with finite characteristic can never be ordered, as we would then have $$1<1+1<1+1+1<\cdots<0$$ –  Alex Becker May 8 '12 at 3:11
I understand that we are trying to prove this by a contradiction. With that i understand why we are letting x be the smallest positive element. What I don't understand is how you show 1/2 is positive and then from that show x/2 is positive. Can you expand a little more please. –  walter miller May 8 '12 at 4:26
@waltermiller: $1$ must be positive, since $1=1^2$; since $1$ is positive, $1+1$ is positive. Since $1+1$ is positive, $\frac{1}{1+1}$ is positive. –  Arturo Magidin May 8 '12 at 5:09