Doubt reagarding denseness of a set in (0,1)

I have been doing a basic math course on Real analysis...I encountered with a problem which follows as" Prove that na(mod1) is dense in (0,1)..where a is an Irrational number , n>=1...

I tried to prove it using only basic principles...first of all i proved that above defined sequence is infinte..and also it is bounded...so by Bolzano-Weierstrass theorem it has a limit point in (0,1)..but to prove denseness i need to prove that for any given (a,b) a subset of (0,1) there is atleast one element of the sequence...Iam not getting how to figure out and link that limit point to that interval (a,b)..can any one help me in this..?...It would be of great help...

OK, so you have proven that the sequence $na$ has a limit point in (0,1). Now, can you prove that for any $\varepsilon>0$, there is an element of the sequence such that $na<\varepsilon$ (that is: can you prove that 0 is a limit point).

Dick