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Assume we have a sequence $(f_n)_{n \in \mathbb{N}}$ of RCLL (i.e. càdlàg, i.e. right-continuous admitting left limits) functions $f_n : [0,T] \rightarrow \mathbb{R}$ with the property that, for every $x \in [0,T]$ there exists $K > 0$ with $\sup_{n \in \mathbb{N}}|f_n(x)| \leq K$. So we can define a function $f^*(x):=\sup_{n \in \mathbb{N}}f_n(x)$ for every $x \in [0,T]$ and see that $f^*:[0,T] \rightarrow \mathbb{R}$ .

My question now is, wheter we can say that $f^*$ is RCLL (i.e. càdlàg), since I cannot think of a counterexample, and cannot come up with a proof.

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The functions $$ f_n(x) = \begin{cases} 0 & x\leq0 \\ nx & x\leq1/n \\ 1 & x\geq1/n \end{cases} $$ are continuous, hence RCLL, but $$ f(x) = \sup_{n} f_n(x) = \begin{cases} 0 & x\leq0 \\ 1 & x > 0 \end{cases} $$ is not right continuous.

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