Is 1/2 less than 1?

Question

Prove that $\frac 12 \lt 1$

We're in the reals and we have the Field Axioms and Order Axiom. I know that for $x \lt y$, $y - x \gt 0$. However I think I'd be assuming what I;m trying to prove if I say that $1 -\frac 12 \gt 0$ and thus $1 \gt \frac 12$. How else can I go about this?

Answer 1

Suppose $1/2 < 0$. Then $1 - 1/2 > 1 - 0 \implies 1/2 > 1 > 0$, a contradiction (if you don't know that $1>0$, prove that a square is always nonnegative and remember $1*1=1$).

Once you have that, then $1/2>0 \implies 1/2=1-1/2<1-0=1$.

Answer 2

2 = 1 + 1 > 1

So 1 = 2/2 > 1/2

Answer 3

If you assume $$~~a > 0 ~~~~ \Rightarrow ~~~~ b + a > b~~\tag A$$ and that $$1/2 + 1/2 = 1 \tag B$$ then:

$$\begin{align}1 & > 1/2 &\text{to prove} \\ 1/2 + 1/2 & > 1/2 &\text{by B}\\ 1/2 &> 0 & \text{by A} \\ &\top &\text{quotient of positives is positive} \end{align}$$