What does the following notation 'sup$_{x\in S}x$' mean in the context of limit superior and inferior? I am asking because usually I see it used like 'sup$X$' for some set $X$, but here it seems as if an element is passed to it.
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I would interpret that as the supremum of the set $S$. I don't know why the author chose to use that notation but it may simply be preference. – CyclotomicField Apr 25 '20 at 13:08
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This is a special case of the fairly common notation $\sup_{\phi(x)}f(x)$ where $f$ is a function and where $\phi(x)$ is a statement about $x$. It means $\sup{f(x):\phi(x)}$. So the expression in your question means $\sup{x:x\in S}$, which is, as @CyclotomicField said, a long-winded way to say $\sup S$. – Andreas Blass Apr 25 '20 at 14:14