This begins my study of sequences and series of functions. These bad boys have a tendency of tripping me up so I’ve decided to spend some quality time learning their ways.
Problem Statement: Let and let be a sequence of real-valued functions on converging uniformly to a bounded real-valued function on . Show there exist constants such that for every when .
Proof: Let be such a sequence converging uniformly to on where is bounded. Since is bounded there exists a such that for every .
Let , then there exists an such that for every since converges to uniformly. Consider for , then it follows that for every . Since can be made arbitrarily small it follows that for every and .
Reflection: This one wasn’t too bad. The key rests in the fact that uniformly, so we may say that for every there exists an such that for and for every, . Since we may say this about every x in we are able to make the bounding argument. If we did not have uniform convergence then we would not be able to make that argument since we would need a different for each .