When I was starting out as a grad student, I remember being frustrated with analysis problems where it didn't use DIRECTLY the things we spent so much time proving in class.
As an example, let's look at a standard real analysis problem:
(1/23)
The problem looks like it can be done using the dominated convergence theorem.
This is an important theorem and is used in a lot of places outside a grad real analysis class. A student would go through a lot of buildup leading to this theorem.
(2/23)
Now, let's see whether the conditions for DCT are satisfied.
First, for fixed 0
0 as n->infinity.
So we have a.e. convergence of the integrand in [0,1].
To the student, this builds a stronger case to use DCT to solve the problem.
(3/23)
Finding an integrable dominating function is where the problem starts.
It would seem NATURAL (at least for me) to bound it with:
p.s. it feels "natural" to me because it is reminiscent of squeeze theorem problems, math students encounter in calculus.
(4/23)
However, 1/x^n is not integrable in (0,1) and hence cannot be (further) dominated by an integrable function.
So, what I would do is since I know it converges a.e. to 1, maybe it converges uniformly to 1?
This leads me into a BAD detour.
(5/23)