Limit of increasing sequence of sets

Definition

Given an increasing sequence of sets $(A_n)_{n=1}^\infty$, we define the limit of the sequence as follows:

• $$\lim_{n\rightarrow\infty}(A_n)=A$$ where $A$ is the limit of $A_n$, and $$A:=\bigcup_{n=1}^\infty A_n$$

This may be written as:

• $A_n\uparrow A$
  • I do not like this notation, as $\uparrow$ only shows the notion of increasing, I prefer $\nearrow$ as this 'combines' (in a very vector-like sense) the $\rightarrow$ of limit and $\uparrow$ of increasing.
• $A_n\nearrow A$
  • I prefer this notation, however I always explicitly write $$\lim_{n\rightarrow\infty}(A_n)=A$$ myself, after letting $(A_n)_{n=1}^\infty$ be an increasing sequence.

Alec's definition

When I encountered this in a book() I didn't read on and formulated myself the definition of $\lim_{n\rightarrow\infty}(A_n)=A$ if:

• $$\forall x\in A\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies x\in A_n]$$ and
• $$\forall n\in\mathbb{N}[A_n\subseteq A]$$

Notice the first one alone is insufficient as any subset of some $A_n$ will satisfy it, so I formulated the second. The first also contains the increasing sequence idea as it requires after a certain index all sets contain $x$.

[Expand]

Proof that

$$\lim_{n\rightarrow\infty}(A_n)=A:=\bigcup_{i=1}^\infty A_n\iff$$ those two conditions.

---

---

TODO: Be bothered, I've done this on some anonymous notepad, it isn't difficult

---