Proof by mathematical induction (nested sets) Mathematical induction works as follows:

*

*Proposition P is true for n=1 (or $n=0$);


*If proposition P is true for a given $n \in \mathbb{N}$, then P is true for $n+1$.
Then P is true for every $n \in \mathbb{N}$.
Now suppose that we want to show by mathematical induction that exists a sequence of non-empty sets:
$C_0,C_1,\dots,C_n,\dots$
such that:
$C_0 \supset C_1 \supset \dots \supset C_n \supset \dots$
This is our proposition. That means that we want to show that:

*

*$\exists \,C_0,C_1 \mid C_0 \supset C_1$;


*If $\exists \,C_0,C_1,\dots,C_n \mid C_0 \supset C_1 \supset \dots \supset C_n$
then $\exists \,C_0,C_1,\dots,C_n,C_{n+1} \mid C_0 \supset C_1 \supset \dots \supset C_n \supset C_{n+1}$.
Is this the correct way to proceed in this case? Thank you!
 A: This would not quite get what you want. By applying induction to 1 and 2 you get for every $n$ a chain $C_0 \supsetneq \ldots \supsetneq C_n$. You just do not get an infinite chain.
Compare it to the following. Let $P(n)$ be the statement there is a sequence of natural numbers $a_1, \ldots, a_n \in \mathbb{N}$ such that $a_1 > a_2 > \ldots > a_n$. It is easy to show that $P(0)$ is true and that $P(n)$ implies $P(n+1)$. So by induction $P(n)$ is indeed true for all $n$. However, we can never get an infinite strictly decreasing sequence
$$
a_1 > a_2 > \ldots > a_n > \ldots
$$
of natural numbers.
So when combining induction with existence statements, you have to be careful. Because you only get existence for each finite (however big) $n$. You do not get existence of the infinite thing. As mentioned in the comments, you could work around this. For example in your decreasing sets case we can define $C_n = \mathbb{N} - \{0, \ldots, n\}$. So we already have an infinite collection of sets. Then by induction we can prove that this is indeed a decreasing chain. Now the induction is no longer applied to prove the existence of something, just to prove a desired property about something that already exists.
A: (Posted after previous answer was accepted)
Let $\{C_n: n\in N\}$ be a family of sets.
Define proposition $P$ such that

$\forall n \in N: [P(n) \iff \forall m \in N: [m<n \implies C_{m+1} \subset C_m]]$.

Then proceed as you would to prove by induction that, $\forall n\in N: P(n)$.
Note that $P(0)$ will vacuously true in this case however you may define the $C_n$
