TITLE: length of an interval as a limit
QUESTION [7 upvotes]: Let $I$ be an interval and $A_{n}$ be the set of $k/n$ where $k$ is an integer.
Prove that the length of $I$ (the positive difference between the end points) is the limit as $n$ tends to infinity of $\frac{1}{n}|(I \cap A_{n})|$.
My plan was to split it up into cases for the different type of intervals and come up with formulas for $|(I \cap A_{n})|$, but I'm finding that very tricky.
Thanks

REPLY [3 votes]: Hint: how different can $|I|$ be from $|IA_n|$ (as a function of $n$)? You should be able to find a bound that goes to $0$ as $n$ gets large