Let
$$f_n(z)=\sum_{k=0}^n(-1)^n\frac{z^{2k}}{(2k)!}\tag{1}$$
$$f(z)=\lim_{n\to\infty}f_n(z)=\cos(z)\tag{2}$$
What does it mean by the following statement?
$f_n(z)$ converges uniformly towards $f(z)$ over compact subsets of $\mathbb{C}$.
In other words, what facts (reasoning) do we need to show first and then to conclude with the statement: Thus $f_n(z)$ converges uniformly towards $f(z)$ over compact subsets of $\mathbb{C}$.