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 $\epsilon > 0$ 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 ($\mathbb C$).
