Principal Ideal Domain fulfills Ascending Chain Condition

Theorem
Let $R$ be a principal ideal domain.

Then $R$ fulfills the ascending chain condition.

Proof
Let $I_1\subseteq I_2\subseteq I_3\subseteq \ldots$ be an ascending chain of ideals.

Build $\displaystyle I = \bigcup_{i \mathop = 1}^\infty I_i$.

$I$ is an ideal.

Since $R$ is a principal ideal domain, $I = \left({a}\right)$ for some $a \in R$.

Now, since $a \in I$, there is some $n$ such that $a \in I_n$.

Thus $\left({a}\right) \subseteq I_n$.

By definition $I_n \subset I = \left({a}\right)$, and so $I_n = I$.

Thus:
 * $\forall m \ge n: I_m = I$