The proof of existence of an infinite sequence of disjoint sets.

Here is the question I am asking about:

Let $\mathcal{M}$ be an infinite $\sigma$-algebra.\a. $\mathcal{M}$ contains an infinite sequence of disjoint sets.\

Here are many links I found here for the solution:

1- An infinite $\sigma$-algebra contains an infinite sequence of disjoint sets

2- Prove that an infinite sigma algebra contains an infinite sequence of disjoint sets and is uncountable

But I am wondering why these two solutions considered the complement case, why we need induction?