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Let $X:= \mathbb R^n, Y:= \mathbb R \cup\{+\infty\}$, and $(u_k)$ with $u_k:X \to Y$ be a sequence of convex functions such that $u_k$ converges pointwise to $u:X \to Y$. The effective domain of $u$ is defined as $\operatorname{dom} u := \{x\in X \mid u(x) \in \mathbb R\}$. Assume that $\operatorname{int} (\operatorname{dom} u) \neq \emptyset$.

Then the author of this post said that

  • I have proved that, for every compact subset $K$ of $\operatorname{dom} u$, the sequence $(u_k)$ is equi-bounded and equi-Lipschitz on $K$.
  • I want to prove that $(u_k)$ converges uniformly on every compact subset of the effective domain of $u$.

First, we fix

  • a compact subset $K$ of $\operatorname{dom} u$,
  • a point $a \in K$,
  • and $N \in \mathbb N^*$.

Assume $u_k(a) = +\infty$ for all $k \le N$. Then the sequence $(u_k)$ can not be equi-bounded and not equi-Lipschitz on $K$. However, the pointwise limit of $(u_k (a))_{k \in \mathbb N}$ is not effected by those $u_k$'s with $k \le N$. So I suspect the author's claim is not correct?

Could you please confirm if my understanding is correct?

Akira
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  • yes, you are right. – daw May 25 '22 at 18:08
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    I added another counterexample to those question. Could you please check? – daw May 25 '22 at 18:13
  • @daw I could not see anything wrong with your example. It may be worth noticing that $(f_n)$ converges uniformly to $0$ on $(-\infty, 1-\varepsilon]$ for any $\varepsilon >0$. Could you post your counter-example as an answer so that I can accept it? – Akira May 25 '22 at 18:43

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