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Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, the sequence $\big(f_n(x)\big)$ is decreasing, and $\lim_{n\to +\infty}f_n(x)=f(x)$.

Question: Do $\lim_{n\to +\infty}\sup_K(f_n)=\sup_K(f)$?

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    $\begingroup$ You might want to look up Dini's theorem. I'm voting to close since this does not appear to be a research level question. $\endgroup$
    – Jochen Glueck
    Commented Jul 4, 2022 at 21:48
  • $\begingroup$ Dini's theorem applies for real-valued functions only. Here, $f_n$ and $f$ have values over $\bar{\mathbb{R}}$. $\endgroup$
    – G. Panel
    Commented Jul 4, 2022 at 21:58
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    $\begingroup$ Dini's theorem also requires the limit function to be continuous. It's rather the proof of the theorem that's interesting for your situation. ;-) $\endgroup$
    – Jochen Glueck
    Commented Jul 4, 2022 at 22:07

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