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Classical Dini's theorem states that if $(f_n)$ is a monotone sequence of continuous functions on a compact space converges pointwise to a continuous function $f$, then the convergence is uniform.

It is shown that all conditions are needed for the conclusion to hold.

If one examines carefully, the proof can be modified to obtain the following statement:

If $(f_n)$ is a monotone sequence of upper semicontinuous functions converges on a compact space converges pointwise to a lower semicontinuous function $f$, then the convergence us uniform.

Question: Suppose for each natural number, $f_n = g_n - h_n$ where $g_n,h_n$ are upper semicontinuous functions. If $(f_n)$ is a decreasing sequence of functions defined above on a compact space converges pointwise to $0$, then is it true that the convergence us uniform?

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  • $\begingroup$ Did you intend your modified statement of Dini to include both increasing and decreasing sequences? $\endgroup$
    – Harald Hanche-Olsen
    Commented May 8, 2017 at 15:29
  • $\begingroup$ No, I just want decreasing sequence. $\endgroup$
    – Idonknow
    Commented May 8, 2017 at 15:38
  • $\begingroup$ I am unsure about the statement as is. I think the statement should be either "sequence of upper semicontinuous + decreasing to a lower semicontinuous limit" or "sequence of lower semicontinuous + increasing to an upper semicontinuous limit". $\endgroup$
    – Just dropped in
    Commented Sep 13, 2021 at 8:37
  • $\begingroup$ My point is, I don't see how to modify the proof in the case that the sequence consists of upper semicontinuous functions that increase to a lower semicontinuous function. $\endgroup$
    – Just dropped in
    Commented Sep 13, 2021 at 8:38

1 Answer 1

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No. Let $f_n\colon[0,1]\to\mathbb{R}$ be the characteristic function of the open interval $(0,1/n)$. It is the difference between $g_n=1$ (continuous) and the characteristic function of $\{0\}\cup[1/n,1]$ (upper semicontinuous).

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