What is the real analysis version of this complex analysis Weierstrass theorem? I'm reading Gong Sheng's Concise Complex Analysis, where it introduced a Weierstrass Theorem

Theorem 3.1 (Weierstrass Theorem) Suppose $\{f_n(z)\}$ is a sequence
of functions where each $f_n(z)$ is defined and holomorphic in a
region  $U\subseteq \mathbb C$. Assume that $\sum_{n=1}^\infty f_n(z)$
converges uniformly to $f(z)$ on every  compact subset of $U$.
Then $f(z)$ is holomorphic on $U$ and for every $k\in \mathbb N$,
$\sum_{n=1}^\infty f_n^{(k)}(z)$ converges uniformly to $f^{(k)}(z)$
on every compact subset of $U$.

Then it mentions:

This is a profound result. The reader can compare it with the theorem
of  the derivative of function series in calculus.

So what is the corresponding real analysis version of this complex analysis Weierstrass Theorem, and what is the difference? -- I suppose the difference would show some distinct properties in complex analysis.
 A: We find in section 10.2.3 Uniformly Convergent Series of Concise Calculus by Sheng Gong the following theorem:

Given a function series $\sum_{n=1}^\infty u_n(x)$ on $[a,b]$. If every term of a function series is differentiable, every derivative function is continuous, and the series of derivative functions is uniformly convergent, then the series is termwise differentiable.
That is, if a functional series of $\sum_{n=1}^\infty u_n(x)$ is convergent, the derivative function $u_n^{\prime}(x)$ of each term is continuous and the function series $\sum_{n=1}^\infty u_n^{\prime}(x)$ is uniformly convergent, then
\begin{align*}
\frac{d}{dx}\left(\sum_{n=1}^\infty u_n(x)\right)=\sum_{n=1}^\infty \frac{d}{dx}u_n(x)=\sum_{n=1}^\infty u_n^{\prime}(x).
\end{align*}
In other words, the order of the derivative $\frac{d}{dx}$ and the sum $\sum$ can be exchanged.

Conclusion: Here we see how strong the concept of a holomorphic function is. In the Weierstrass theorem we consider functions $f_n$ being holomorphic on a compactum. This implies also the existence of derivatives of the functions $f_n$ of arbitrary order.

*

*Armed with that it is sufficient to require uniform convergence of the holomorphic functions $f_n$ on a compactum and it follows not only that $f(z)=\sum_{n=1}^\infty f_n(z)$ is holomorphic on this compactum but also the uniform convergence of all $k$-th derivatives $f^{(k)}$ of $f$ on it.


*On the other hand, in the real case we have to require that all the functions $u_n(x)$ are $C^1$, i.e. differentiable and the derivatives are continuous. Contrary to the complex case we have to require that the derivatives are uniformly convergent on $[a,b]$  and we cannot conclude anything for higher derivatives of the functions $u_n(x)$, since we even don't know if they exist.
