Let $\displaystyle I_{1} \subseteq I_{2} \subseteq I_{3} \ldots$ be an increasing sequence of ideals in a ring R

1.) Prove that $\displaystyle \bigcup_{i\in\aleph} I_{i} $ is an ideal.

2.) Prove that if every ideals of R is finitely generated, then exist a number $\displaystyle n \in \aleph $ such that

$\displaystyle I_{1} \subseteq I_{2} \ldots \subseteq I_{n-1}\subseteq I_{n} = I_{n} = I_{n+1} =I_{n+2} = \ldots

$