1
$\begingroup$

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??

$\endgroup$
1
$\begingroup$

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 $1-1=0<nx$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.