The following Theorem as stated is from Rudin's Principles of Mathematical Analysis.

7.25 Theorem If $K$ is compact, if $f_n \in \mathscr{C}(K)$ for $n = 1,2,3,...,$ and if $\{f_n\}$ is pointwise bounded and equicontinuous on $K$, then

(a) $\{f_n\}$ is uniformly bounded on $K$,

(b) $\{f_n\}$ contains a uniformly convergent subsequence.

Question If $\{f_n\}$ is a family of equicontinuous functions on a compact $K$, then doesn't that automatically imply $f_n \in \mathscr{C}(K)$, where $\mathscr{C}(K)$ is the set of complex-valued, continuous, bounded functions? If so, why did Rudin include the $``$if $f_n \in \mathscr{C}(K)$ for $n = 1, 2, 3, ...,$'' part in his hypothesis? Is there something I am missing?

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    $\begingroup$ It's redundant, sure, but in a sense the hypothesis about $f_n$ is sort of specifying the type of objects we're looking at, and the rest is specifying what they do. This is a common situation in mathematics. If you didn't write that, you'd still want to say "$f_n : K \to \mathbb{C}$" (or "$f_n$ are real-valued functions on $K$", or similar) and it would take even more symbols to write out anyway. $\endgroup$ – Ian Mar 22 '17 at 18:33

Logically it is redundant, but it makes sense to say it this way.

Equicontinuity is a property of families of continuous functions so semantically it makes more sense to establish first that you have a family of continuous functions and then introduce the assumption that they are equicontinuous.

I personally don't like repeated "if"s, so I would say something more like

Let $K$ be a compact set and $\{f_n\}_{n=1}^{\infty} \in C(K)$. If $\{f_n\}_{n=1}^{\infty}$ is equicontinuous, then .....

So that you only have a clear single, "if.., then..." statement.


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