Increasing Sequence of Sets induces Partition on Limit

Theorem
Let $\sequence {S_n}_{n \mathop \in \N} \uparrow S$ be an increasing sequence of sets with limit $S$.

Define $T_1 = S_1$, and, for $n \in \N$, $T_{n + 1} = S_{n + 1} \setminus S_n$, where $\setminus$ denotes set difference.

Then $\sequence {T_n}_{n \mathop \in \N}$ is a countable partition of $S$.

Proof
That $\sequence {T_n}_{n \mathop \in \N}$ partitions $S$, means precisely that:


 * $(1): \quad$ The $T_n$ are pairwise disjoint
 * $(2): \quad \ds \bigcup_{n \mathop \in \N} T_n = S$

It is more convenient to prove $(1)$ and $(2)$ separately:

Proof of $(1)$
Let $l, m \in \N$ be such that $l < m$.

Then by Set Difference is Subset, $T_l \subseteq S_l$.

As the $S_n$ form an increasing sequence of sets, it follows that also $T_l \subseteq S_{m - 1}$ because $m - 1 \ge l$.

Now compute as follows:

Hence $T_m \cap T_l = \O$.

Reversing the roles of $m$ and $l$ leads to the same conclusion if $l > m$.

Hence, by definition, the $T_n$ are pairwise disjoint.

Proof of $(2)$
By Set Union Preserves Subsets and Set Difference is Subset, have that:


 * $\ds \bigcup_{n \mathop \in \N} T_n \subseteq \bigcup_{n \mathop \in \N} S_n = S$

To establish $(2)$, by definition of set equality, it is now only required to show that $\ds S \subseteq \bigcup_{n \mathop \in \N} T_n$.

So let $s \in S$.

Then by definition of union, the set:


 * $N_s := \set {n \in \N: s \in S_n}$

is nonempty.

By Well-Ordering Principle, $N_s$ contains a minimal element, $n$, say.

If $n = 1$, then $s \in S_1 = T_1$.

If $n > 1$, then, by minimality of $n$, $s \notin S_{n - 1}$.

Hence, by definition of set difference, $s \in T_n = S_n \setminus S_{n - 1}$.

By definition of set union, it follows that:
 * $\ds s \in \bigcup_{n \mathop \in \N} T_n$

That is, by definition of subset:
 * $\ds S \subseteq \bigcup_{n \mathop \in \N} T_n$