# Math Help - analysis

1. ## analysis

Any ideas on this problem? Thanks!

Let a be a positive number. Prove that for each real number x there is an integer n such that na is less than or equal to x is less than (n+1)a.

2. Originally Posted by friday616
Any ideas on this problem? Thanks!

Let a be a positive number. Prove that for each real number x there is an integer n such that na is less than or equal to x is less than (n+1)a.
You know that there is an integer $n$ such that $na>x$ - by the Archimedean property. Consider the set $S=\{ n | na \leq x \}$. This set must be bounded by what we have just set above. Let $N = \max\{ S\}$. Then we have that $Na \leq x$, but by construction $N+1\not \in S$ since it is larger than the maximum and so $(N+1)a > x$.