I'm reading the third addition of Rudin's "Principles of Mathematical Analysis".
There's a very small detail in the proof of Theorem 1.20 on page 9 that confuses me a little. I feel like I'm missing something. The Theorem proves the fact that between any two real number $x<y$, there is a rational number $p$ with $x < p < y$.
He uses the Archimedean Property (for any two real numbers $x$ and $y$, with $x>0$, there exists a positive integer $n$ or which $y < nx$) to prove that for any real number $w$, there are positive integers $m_1$ and $m_2$ for which $-m_1 < w < m_2$.
This is what confuses me: Having taken such care to prove the self-evident fact that there exist $m_i$ for which $-m_1 < w < m_2$, he concludes, with no further justification that there exists an integer $m$ with $-m_1 \le m \le m_2$ for which $m-1 \le w < m$.
Again, it seems self-evident that for any real number $w$, there exists an integer $m$ for which $m-1 \le w < m$. But how can he brush-over this when he insisted on proving $-m_1 < w < m_2$?