I need help on how to prove this:
If x is an arbitrary real number, prove that there are integers m and n such that m < x < n.
Thank you
WLOG for any consider . By the Archimedean Ordering on there shall exist such that . Hence we can find . That completes the second inequality.
For any consider . I claim we can choose an integer such that .
Above we can find an integer . Hence consider the interval . Define the set of integers . This is a finite non-empty set. So choose . I leave it to you to prove that is the desired integer.