TITLE: Prove that there is no smallest positive real number
QUESTION [11 upvotes]: I have to prove the following: 
$$\text{Prove that there is no smallest positive real number}$$
Argument by contradiction
Suppose there is a smallest positive real number. Let $x$ be the smallest positive real number: 
$$x : x \gt 0, x \in \mathbb{R}$$
Let $y$ be $\frac{x}{10}$. Contradiction. This implies that $y < x$ which implies that you can always construct a number that is less than the "smallest positive real number". QED. 
Can someone please verify the write up of the proof and the proof itself? 
Thanks for your time!
P.S. I have seen this and this but I'm not looking for a way to approach the problem but rather verification and write up help.
P.P.S If there is another novel way of approaching this problem, I would like to know!

REPLY [3 votes]: If $x$ is the smallest real,
then $0 < x < 1$.
But 
$1-x > 0$ so
$0 < x(1-x)
=x-x^2$
so $x^2 < x$.
(Just wanted to come up with a different answer.)