Krdytkyu

Theorem 7.8 The sequence of functions ${f_n}$ defined on $E$ converges uniformly on $E$ if and only if for every $epsilon > 0$ there exists an integer $N$ such that $m geq N, n geq N, x in E$ implies
begin{equation}
|f_n(x)-f_m(x)| leq epsilon
end{equation}

For the backwards direction, since the codomain of $f$ is not given, how can we use Theorem 3.11 (Cauchy sequence in a compact metric space (or $mathbb{R}^k$) converges to some point in the metric space) to prove pointwise convergence of $f$?

For each $x in E$, the sequence $(f_n(x))_{n in mathbb N}$ is a Cauchy-sequence in $ mathbb R$ or ($mathbb C$).

From Chapter 1:

1.38 Remarks Theorem 1.37 (a), (b), and (f) will allow us (see Chap. 2) to regard $R^k$ as a metric space. $R^1$ (the set of all real numbers) is usually called the line, or the real line. Likewise, $R^2$ is called the plane, or the complex plane (compare Definitions 1.24 and 1.36). In these two cases the norm is just the absolute value of the corresponding real or complex number.

From Chapter 2:

2.16 Examples The most important examples of metric spaces, from our standpoint, are the euclidean spaces $R^k$, especially $R^1$ (the real line) and $R^2$ (the complex plane); the distance in $R^k$ is defined by

$text{(19)} quad quad d(x. y) = |x — y| ; ; ; text{ with } x, y in R^k$

By Theorem 1.37. the conditions of Definition 2.15 are satisfied by (19).

If you were reading Rudin in self-study mode, you might have to use the index to find complex plane. I tried to put myself in the OP's shoes and saw the term right under complex number.

But, it does seem unfair for Rudin to bury theory inside remarks and examples.

It is a bit humorous how in the remark Rudin writes

$R^1$ (the set of all real numbers) is usually called the line, or the real line.

and in the next sentence states

Likewise, $R^2$ is called the plane, or the complex plane

Apparently the definition of complex plane can be found in a remark that it will all make sense in the next chapter. And if we call $R^2$ a plane then that means we are examining Euclidean 2-space.

Rudin wasn't using latex, representing the real numbers with $R$ while not using any symbol for the complex plane; see the LIST OF SPECIAL SYMBOLS at the end of the book

I cannot quickly scan the whole text to check all references to 7.8. Based on the contents of chapter Uniform convergence and continuity (pages 149-151), Theorem 7.8 is used to prove Theorem 7.15, which is essentially about complex-valued, bounded and continuous functions $mathcal{C}(X)$.

Theorem 3.11 is about metric spaces that are either compact, $mathcal{R}^n$ or $mathcal{C}$.

So $E$ in Theorem 7.8 is a typo. Actually, it should be $mathcal{C}$ as we are dealing with complex-valued functions.

Yes, this book had 3 editions, so many views, but no mention of this bit in errata.

Errata from George Bergman does not include this error https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_notes.pdf

Notes from Jiří Lebl point to complex-valued functions for Theorem 7.8, page 3 of 21 in PDF
https://math.okstate.edu/people/lebl/uw522-s12/lec1.pdf