1. So now we know that for a sequence to be convergent it must be montonic and bounded. Therefore . So for every point where converges we have by neccessity . And if then . Thus on is bounded.
2. Before we start the next step we are given that is Riemann integrable, thus bounded and continous.
3.Next we just use the theorem that if and is continuous on then is continous on . So the fact that tells us that is continuous on
4. So we have shown that on is both bounded and continous. So it is Riemann integrable.