I need to prove that if the sequence of functions
converges uniformly to on , and each is integrable, then the limit function is also integrable on [0,1].
Does this help at all??
Since converges uniformly to , then
such that
I need to prove that if the sequence of functions
converges uniformly to on , and each is integrable, then the limit function is also integrable on [0,1].
Does this help at all??
Since converges uniformly to , then
such that
First let us remark that where represents the th partial sum. So now
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.