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](https://math.stackexchange.com/questions/959863/prove-using-a-proof-by-contradiction-there-is-no-smallest-positive-real-number) and [this](https://math.stackexchange.com/questions/1171640/prove-there-is-no-smallest-positive-rational-number) 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!