Suppose that each function in a sequence $\{f_n\}$ is continuous in an open set $U$ and that the sequence converges normally in $U$ to the limit... I am reading the book of Palka, An Introduction to Complex Function Theory, and the author on page 248  states two theorems, which does not prove and I would like to know what the demonstration is. The topics are the following:
Theorem $1.4.$ Suppose that each function in a sequence $\{f_n\}$ is continuous in an open set $U$ and that the sequence converges normally in $U$ to the limit function $f$. Then $f$ is continuous in $U$. Furthermore $\int_{\gamma}f(z)dz=\lim_{n\rightarrow \infty }\int_{\gamma}f_n(z)dz$ for every piecewise  smooth path $\gamma$ in $U$. 
Theorem $1.5.$ A sequence $\{f_n\}$ of functions defined in on open set $U$ converges normally in $U$ if and only if it is a uniform Cauchy sequence on each closed  disk that is contained in $U$.
I do not understand why Palka does not show these results, could someone explain to me please, I think that to solve this one can use one of the following results:

 A: Theorem 1.4: Let $z_0\in U$ and consider a $r>0$ such that the closed disk $\overline{D(z_0,r)}\subset U$. Then $\overline{D(z_0,r)}$ is a compact set and therefore the restriction of $(f_n)_{n\in\mathbb N}$ converges uniformly to $f$ on $\overline{D(z_0,r)}$. Therefore, by theorem 1.1, $f$ is continuous at $z_0$.
Besides, since the image $\Gamma$ of $\gamma$ is compact and $(f_n)_{n\in\mathbb N}$ converges uniformly to $f$ on $\Gamma$, theorem 1.1 tells us that$$\lim_{n\to\infty}\int_\gamma f_n(z)\,\mathrm dz=\int_\gamma f(z)\,\mathrm dz.$$
Theorem 1.5: Since closed disks are compact sets, every normally convergent sequence converges uniformly on such disks and therefore it is a uniform Cauchy sequence there.
On the other hand, if a sequence $(f_n)_{n\in\mathbb N}$ of functions is a uniformly Cauchy sequence on every closed disk and if $K$ is a arbitrary compact subset of U, one can use the compacity of $K$ to find a finite set $z_1,z_2,\ldots,z_n$ of elements of $U$ and a finite set $r_1,r_2,\ldots,r_n$ of numbers greater than $0$ such that each closed disk $\overline{D(z_k,r_k)}$ is a subset of $U$ and that$$K\subset\bigcup_{k=1}^n\overline{D(z_k,r_k)}.$$Since $(f_n)_{n\in\mathbb N}$ is a uniformly Cauchy sequence on each such disk, it is a uniformly Cauchy sequence on $K$.
