We were given a set $ A \subset \mathbb R$ that is compact and a sequence of functions $f_n$ that is point-wise convergent for all $x \in A $. The sequence is monotonically decreasing and it converges to a continuous $ f : A \rightarrow \mathbb R$.
The question is the following: If every element of the sequence $f_n$ is upper semi-continuous, is the sequence uniformly convergent?
I thought that if the sequence of functions is pointwise convergent, and the functions are continuous, it implies that it is as well uniformly convergent, but I'm not sure on how to expand that to the case of semi-convergence.
What I'm also confused about is if it is also valid for lower semi-continuous functions in a sequence?
Any tips on how to approach the problem are welcome!
If you go through the proof that you find e.g. in Baby Rudin, Thm. 7.13, you see that the point is that, for any fixed $\epsilon > 0$, the set $$ K_n := \{x\in A:\ f_n(x) - f(x) \geq \epsilon\} $$ is closed (hence compact, being $A$ compact). This fact is still true if $f_n$ is u.s.c. and $f$ is continuous (or, at least, l.s.c.).
If your sequence is increasing (instead of decreasing) and converges pointwise to a continuous $f$, then the same result holds true provided that the $f_n$ are l.s.c.