Let $\Omega$ be a nonempty set and $\{A_{i}\}_{i\in\mathbb{N}}$ be a sequence of subsets of $\Omega$ such that $A_{i+1}\subset A_{i}$ for all $i\in\mathbb{N}$. Verify that $\mathcal{A} = \{A_{i}:i\in\mathbb{N}\}$ is a $\pi$-system and also determine $\lambda\langle\mathcal{A}\rangle$, the $\lambda$-system generated by $\mathcal{A}$.
MY ATTEMPT
Let $A\in\mathcal{A}$ and $B\in\mathcal{A}$. Then either $A\subseteq B$ or $B\subseteq A$.
Without loss of generality, let us assume that $A\subseteq B$. Then we have that $A\cap B = A\in\mathcal{A}$. Thus $\mathcal{A}$ is a $\pi$-system.
Since $\mathcal{A}$ is a $\pi$-system, one has that $\lambda\langle\mathcal{A}\rangle = \sigma\langle\mathcal{A}\rangle$, the $\sigma$-algebra generated by $\mathcal{A}$.
But I do not know to proceed from here. Can someone help me with this?
As you noted, $\lambda(\mathcal A)=\sigma(\mathcal A)$. First define $\mathcal B=\{A_1,A_2\cap A_1^c,A_3\cap A_2^c,...\}$ and (exercise!) $\sigma(\mathcal A)=\sigma(\mathcal B)$ so that $\lambda(\mathcal A)=\sigma(\mathcal B)$. Now $\sigma(\mathcal B)$ is just going to be the set of all possible countable unions of sets of the type $(\bigcup A_n)^c\cup A_i\cap A_{i-1}^c$ or $A_i\cap A_{i-1}^c$ since $\mathcal B$ is a set of partitioning sets for $\Omega$.