Can we prove $\frac{1}{2}$ is positive? It is well known that we can prove $1>0$ using trichotomy properties of $\mathbb{R}$. Can we prove $\frac{1}{2}>0$ using trichotomy properties of $\mathbb{R}$? Any help would be appreciated. Thank you

Here is a roundabout proof.

First prove that any nonzero real number squared is positive. This can be shown easily by taking two cases: squaring a positive number and showing it's positive, and squaring a negative number and showing it's positive.

Once you have done this, note that $1/4 = (1/2)^2$ so we know that $1/4 > 0$

Also see that $1/2 = 1/4 + 1/4$; that is, $1/2$ is the sum of two positive numbers, which must also be positive