Help with understanding a sequence.

[Pat Says]

Ok, so we are given that {an} is the sequence

 

1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 2/6, ....

 

Now, the problem says suppose that 0< a < b <1. Let N(n; a,b) be the number of integers j<n such that aj is in [a,b].

 

I can't figure out exactly what this means ( i.e. given any n, a, b what does it equal?)

 

It does say that "Thus N(2; 1/3, 2/3) = 2, and N(4; 1/3, 2/3) = 3."

 

Our actual job is to prove that lim n->infinity of [N(n; a,b)]/[n] = b-a.

 

So, I can't begin to start the problem until I figure out what the question even means! Also, if the value is supposed to be an integer then how can the limit be b-a (a non integer.. for the most part)? By the way, I am assuming that n is supposed to be a natural number as it usually is in this book and seems to also be so in this context.

 

If someone could just help explain how to figure out the value of any given a,b,n it would be greatly appreciated and at least give me something to work with.

 

Thank you to everyone in advance.

 

Note (in case it helps): On part b of the question it says that a sequence {an} of numbers in [0,1] is called uniformly distributed in [0,1], if the limit above holds.