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?
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 ...