1. Let a be a positive number. Prove that for each positive real number x there is an integer n such that na≤x≤(n+1)a. I have been looking through mounds of books, but haven't figured out where to start. Our teacher just left us hanging on how to figure it out. I am severely stuck and need help on how to get the problem started. Thanks.
What do you have to work with? Do you, for example, know that "for any y, there exist n such that [itex]n\le y\le n+1[/itex]"? If so, think about y/a. If you don't know that, do you know that "every set of non-negative integers has a smallest member" (the "well-ordered property" of the non-negative integers). It's not too difficult to use that to prove the property I mentioned above. Think about the set of all positive integers larger than y. Obviously, in order to prove something about positive integers, you have to use some property of positive integers!