# question about existence of a positive integer

Let $x \in \mathbb{R}$ such that $x > 0$. Say we are given positive integers $n,m_1,m_2$ such that

$$-m_2 < nx < m_1$$

Question: I am having really hard time trying to see why there must exist a positive integer $m$ with $-m_2 \leq m \leq m_1$ such that

$$m-1 \leq nx < m$$

Why is this true??

Consider the set $\{m_1, m_1-1, m_1-2,\ldots, 1\}$.
1. We have $m_1>nx$. If $m_1-1\le nx$, we stop, otherwise we continue to step 2.
2. We have $m_1-1>nx$. If $m_1-2\le nx$, we stop, otherwise we continue to step 3.
3. We have $m_1-2>nx$. If $m-1-3\le nx$, we stop, otherwise we continue to step 4.
This finite algorithm must terminate, since $1-1=0<nx$.