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

$I$ is an ideal.

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

Now, since $a\in I$, there is some $n$ such that $a\in I_n$. And then $(a)\subseteq I_n$.

By definition $I_n\subset I=(a)$, and so $I_n=I$.

Thus, $\forall m\ge n,\ I_m=I$