Archimedean Property Corollary proof I am trying to prove one of the Archmedian properties corollaries.



The corollary states:

If $x$ is a real number such that $x&gt;0$ then there exists a natural number $n\in \mathbb{N}$ such that $n−1\le x &lt;n$.



From the Archmedian property, I know that for $x\in \mathbb{R}$, then there is a positive integer $n$ such that $n&gt;x$ and from the Well Ordering Principle I know that for any nonempty subset of $\mathbb{N}$ there is a least element, however, how do I show that $n-1\le x$?
look at all those natural numbers $n$ which is larger than $x$, and choose the smallest one (e.g. $n_0$) by well-ordering principle, then argue that $n_0-1$ is also a natural number(shall make use of the fact that $x&gt;0$). Hence it must follow that $n_0-1\le x$.
look at all those natural numbers $n$ which is larger than $x$, and choose the smallest one (e.g. $n_0$) by well-ordering principle, then argue that $n_0-1$ is also a natural number(shall make use of the fact that $x&gt;0$). Hence it must follow that $n_0-1\le x$.