A Type of Union I do not understand

Question

I am not able to read this following union. Could someone please help me to understand it?

$\bigcup\limits_ {k \geq n}${${x \in X: |f_k(x)-f(x)|\geq q} $ }

Answer 1

Let $E_k = \{x \in X: \vert f_k(x) - f(x) \vert \geq q\}$ for all $k \geq 1$.The set you ask is simply $$\bigcup_{k \geq n} E_k$$ Note that $x \in \bigcup_{k \geq n} E_k$ if and only if $x \in E_k$ for some $k \geq n$ if and only if $$\vert f_k(x) - f(x) \vert \geq q \text{ for some } k \geq n$$ Therefore the set $\bigcup_{k \geq n} E_k$ is simply $$\bigcup_{k \geq n} E_k = \{x \in X: \vert f_k(x) - f(x) \vert \geq q \text{ for some } k \geq n\}$$