I am wondering whether the following claim holds or not:
If a sequence of real-valued upper semicontinuous functions is a decreasing sequence and converges to $0$ pointwise, then the convergence is uniform.
This doesn't look true to me, as the answer given by David Mitra suggests the other way. https://math.stackexchange.com/questions/82766/dinis-theorem-and-tests-for-uniform-convergence#=
The result is true if the domain is a compact topological space. The proof is the same as for Dini's theorem, since the only property of continuous functions used in the proof is that $\{x:f(x)<\epsilon\}$ is open, which holds also if $f$ is upper semicontinuous.
If the domain is not compact, the result may fail even if the sequence is made up of continuous functions. Consider for instance $f_n(x)=\max(|x|-n,0)$ on $\Bbb R$.