Math Help - Integrable function

1. Integrable function

Could someone give me a hand on this problem? I want to show that a bounded function $f: [a,b] \rightarrow R$ having a convergent sequence of discontinuous points is Riemann integrable.
I was able to show that that a function bounded function having finitely many discontinuous points is Riemann integrable.
I think I need to come up with a partition P such that $U(P,f)-L(P,f) < \epsilon$ for this problem.

2. If you can do a similar problem for a finite collection of discontinuities, then this problem requires almost the same proof. Because the sequence of discontinuities converges, almost all of the discontinuities are in a cell of any partition containing the limit point as an interior point.