Show that
converges to some real number.
The problem hints that I'm supposed to use the fact that
converges uniformly on [0,1], which I've already proved.
I can't figure out how to connect these two facts at all... any help?
Let's use your hint.
Because of the uniform convergence, the sum is continuous on . Furthermore, you may integrate term by term on . Note also that . We get:
.
The important thing is that the right-hand side is finite.
The above shows (means) that has a finite limit as .
However, this partial sum can be rewritten as ...