I'm trying to prove ; that is, that is Lebesgue integrable over .
Let and .
Since , is an increasing sequence of functions which converges everywhere to .
We want to show that is integrable so it suffices to show converges as . It then follows from the Monotone Convergence Theorem that (and ).
We use the given hint ("don't try to find ; integrate a simpler upper bound instead").
We see that is bounded by ___. The integral of this upper bound converges, so by comparison test converges.
My hurdle is filling the gap for an upper bound. is bounded by the constant function 1 but which doesn't converge as . Any suggestions?