Prove that there is no positive real

1. Mar 26, 2014

cragar

1. The problem statement, all variables and given/known data
Prove that there exists no smallest positive real number.
3. The attempt at a solution
Lets assume for contradiction that x is the smallest positive real.
Now we will look at the midpoint between 0 and x which is x/2, well x/2 is positive and smaller than x so this is a contradiction, so there is no smallest positive real.

2. Mar 26, 2014

jbunniii

Looks fine to me. If you want to be pedantic, you may want to prove explicitly that $x/2$ is positive and strictly less than $x$.

3. Mar 26, 2014

micromass

Staff Emeritus
Depending on how much rigor you need to put in, you might want to prove $0<\frac{x}{2}<x$. I know it's trivial, so perhaps it's not needed.

4. Mar 26, 2014

cragar

ok thanks, it almost seemed to easy just wanted to make sure it worked.