I am studying the proof of existence of least upper bound, but I can not understand how the autor applies the Archimedean property.
Archimedean property. Let $x$ and $\epsilon$ be any positive real numbers. Then there exists a positive integer $M$ such that $M \epsilon > x$.
So, in the proof of existence of least upper bound
Let $E$ be a non-empty subset of $\mathbb R$ with an upper bound $M$, we have to show that $E$ has at least one least upper bound. Since $E$ is non-empty, we can choose some element $x$ in $E$. Let $n \ge 1$. By the Archimedean property, we can find an integer $K$ such that $K/n \ge M$, and hence $K/n$ is also an upper bound for $E$. By the Archimedean property again, there exists another integer $L$ such that $L/n < x$. Since $x$ lies in $E$, we see that $L/n$ is not an upper bound for $E$. Since $K/n$ is an upper bound but $L/n$ is not, we see that $K \ge L$.
I understand the rest of the proof but the part "there exists another integer $L$ such that $L/n < x$" it is no clear for me.