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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 May 8 '17 at 15:29
  • $\begingroup$ No, I just want decreasing sequence. $\endgroup$ – Idonknow May 8 '17 at 15:38
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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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