Do we assume $f_n$'s map into $\Bbb{R}$ or $\Bbb{C}$ in Theorem 7.8 Rudin's *Principles of Mathematical Analysis*? Theorem 7.8 The sequence of functions $\{f_n\}$ defined on $E$ converges uniformly on $E$ if and only if for every $\epsilon &gt; 0$ 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 ($\mathbb C$).
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