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How can I use the Archimedean principle to show that there exists an integer $n$ such that $\frac{1}{n}<y-x$ where $x$ and $y$ are real numbers with $0\leq x<y$?

According to the Archimedean principle: Let $a>0$ and $b \in \mathbb{R}$. There exists an integer $n \in \mathbb{N}$ such that $b<a\cdot n$.

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  • $\begingroup$ How you express the Archimedean principle ? $\endgroup$ – Mauro ALLEGRANZA Feb 5 at 15:26
  • $\begingroup$ @MauroALLEGRANZA I added the description in the question now. $\endgroup$ – Jake Feb 5 at 15:28
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You apply the definition. Because $x \lt y, y-x \gt 0$. Let $y-x$ be the $a$ in the definition and $b$ be $1$.

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