If we consider a family of functions to be a sequence of functions, do we need to require in the definition that $$|f_n(x)-f_n(y)|<\epsilon$$ holds for $n$ large? Or should it hold for all $n$?
The general definition implies that it should hold for all $n$. But I found two implicit answers in Rudin which contradict each other. The proof of Theorem 7.24 implies that the above should hold for $n$ large, where as the proof of Theorem 7.25 says it should hold for all $n$.
On
In 7.24 Theorem, the equicontinuity of $\{f_n\}$ is the part of conclusion. So, Rudin proves it by using uniform continuity in conjunction with uniform convergence. Using uniform continuity, he proves that $|f_{i}(x)-f_{i}(y)| < \varepsilon$ for all $1 \leq i \leq N$, and using uniform convergence, he proves that $|f_{i}(x)-f_{i}(y)| < \varepsilon$ for all $i>N$.
However, in 7.25 Theorem, the equicontinuity of $\{f_n\}$ is a given hypothesis. So, Rudin uses the fact that $|f_{n}(x)-f_{n}(y)| < \varepsilon$ for all $n$ directly.
If you read correctly, Rudin mentioned that $|f_{i}(x)-f_{i}(y)| < \epsilon$ for all $1 \leq i \leq N$. He then showed the same thing for all $i > N$. So yes, it does have to hold for all $n$.