A basic question on limit of a sequence of "events"

A basic question on limit of a sequence of "events"

Suppose we have an increasing sequence of event $A_n$, then is the
following a "definition" or needs "proof"
$$\lim_{n\to\infty}A_n = \bigcup_{n=1}^{\infty}A_n$$
If it is definition how to prove that it is "well-defined" and if it needs
proof what is the proof ? Confused ...