Decreasing sequence of sets: Power of natural numbers

Question

Let $P(N)$ be the set of all the possible subsets of natural numbers (power set of $N$). Suppose that we have a decreasing sequence of sets $S_n$, ie $S_{n+1} \subseteq S_n\;,\in P(N)$ such that they are all finite and a set $M$ such that $\#M \leq \#S_n$, for all $n$. It is possible to say that $$\#M \leq \#\cap_{n=0}^\infty S_n$$?

My intuition says it does, but I couldnt prove it. It seems that the intersection must be one of the $S_n$. Any hint of what should I do?

Thanks!

Answer 1

Let $S$ denote the intersection of the $S_n$. Start with a fixed $m$.

The set $S_m-S$ is finite and for every element $s\in S_m-S$ some $n_s$ exists with $s\notin S_{n_s}$.

Then $S_n=S$ if $$n\geq\max\{n_s|s\in S_m-S\}\in\mathbb N$$