Counterexample for convergence in L1

Suppose that and converges uniformly to f [/tex], by a theorem that I know, and provided that , X being the domain set.

Now, if , this may not be true.

Counterexample:

Define

And define

And uniformly, but is integrable for all n while f is not.

Question:

Q1: Why is uniformly? The book just claim it is, but I'm trying to prove it using the defintion of such that

Q2: I understand why f is not integrable since it goes to infinity, but are integrable? Is it because is finite?

Thank you!