Proof of Archimedean property of $\mathbb{R}$ I have been trying to prove Archimedean property ($n\cdot x>y$) on my own and came up with this.
No matter how big $y$ is given, I can get next integer of that value of $y$. And no matter how small $x$ is, I can get a terminating decimal in base 10, which is rational and then I will make my $n$ as denominator of that rational times the next integer of $y$, which will give $n\cdot x>y$. Is this valid or have I missed something??

I think you are assuming the theorem in your proof, more or less. Regardless you are not showing how you get the rational number you use in your proof.
Assuming you are working with $\mathbb{R}$ as $\mathbb{Q}$ with supremum for nonempty sets bounded above, suppose no such integer exists. Thus $\Bbb Z$ is bounded above by $\frac{y}{x}$, and then has a supremum $s$. It follows that since $s-1$ is less than the supremum of $\Bbb Z$, there exists some $n\in \Bbb Z$ such that $s-1< n$, and therefore $s< n+1$, a contradiction.
