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A sequence of functions $(f_n)$ converges to $f$ uniformly on $S$ if and only if $$ \lim_{n\to\infty}\sup\{|f_n(x)-f(x)|:x\in S\}=0\ (1). $$

I understand this theorem. However, I wondered what the following would mean: $$ \lim_{n\to\infty}\{\sup(|f_n(x)-f(x)|):x\in S\}=0\ (2). $$ Is it equivalent? If not, what is the difference then?

Could you say that in the second expression, you are basically always considering a singleton, while in the first expression, we do actually consider a set? However, I still don't see why the second expression would mean something different.

Or is it that we can't talk about the supremum of a point? It makes no sense to talk about the supremum of $f_n(x_0)-f(x_0)$, while it makes sense to first consider all $x\in S$, and thén take the supremum. This would mean that equation (2) basically is equal to $$ \lim_{n\to\infty}\{|f_n(x)-f(x)|:x\in S\}=0, $$ so without the supremum.

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    Limits are not defined for *sets*. The limit of the original theorem is actually the limit of *the supremum of a set* and hence the limit of a sequence of real numbers. – 2017-02-05
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    @YuxiaoXie Actually, limits are defined for sets, but the definition is different than the one for limits of real numbers. But I understand the difference between both notations now. – 2017-02-05
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    I've only heard of the $\limsup$ and $\liminf$ of a sequence of sets. How is the limit of an arbitrary sequence of sets defined? – 2017-02-05
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    @YuxiaoXie I hope it applies to your question, but a short while ago I posted a question asking about the difference between the limits of sets and the limits of the reals: http://math.stackexchange.com/questions/2108084/difference-between-ordinary-limits-and-limits-of-sets – 2017-02-05

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