= Solution
Let $J=\bigcup_nI_n$. It contains <zero>. If $a,b\in J$, both belong to $I_N$ for some common $N$, so $a-b\in I_N\subseteq J$. If $r\in R$, then $ra\in I_N\subseteq J$. These are the <ideal> axioms, proving $J$ is an <ideal>.
In a <principal ideal domain> write $J=(g)$. Since $g\in J$, there is $n_0$ with $g\in I_{n_0}$. For $n\geq n_0$,
$$
(g)\subseteq I_{n_0}\subseteq I_n\subseteq J=(g).
$$
All inclusions are equalities. Hence \b[the chain stabilizes at $I_{n_0}$]. This includes $J=(0)$ and is the <ascending chain condition> for <ideals>.
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