Proofs
Proof using dominated convergence theorem and assuming that function is bounded
Suppose first that is bounded, i.e. . A change of variable in the integral
shows that
- .
Since is bounded, the Dominated Convergence Theorem implies that
Proof using elementary calculus and assuming that function is bounded
Of course we don't really need DCT here, one can give a very simple proof using only elementary calculus:
Start by choosing so that , and then
note that uniformly for .
Generalizing to non-bounded functions that have exponential order
The theorem assuming just that follows from the theorem for bounded :
Define . Then is bounded, so we've shown that .
But and , so
since .