# Thread: A proof for the union of a family of sets..

1. ## A proof for the union of a family of sets..

• What is $\displaystyle \bigcup_{n \in N} A_n$ where $\displaystyle A_n=\{n, n+1, n+2,..., 2n\}$ ?
I want to say that the union is $\displaystyle \mathbb{N}$, could I prove this by induction? What would be the simplest approach?

2. The simplest approach is to note that for every $\displaystyle n\in\mathbb{N}$, $\displaystyle n\in A_n$, so $\displaystyle n\in\bigcup_{n\in\mathbb{N}}A_n$. (And, obviously, $\displaystyle \bigcup_{n\in\mathbb{N}}A_n\subseteq\mathbb{N}$ because each $\displaystyle A_n\subseteq\mathbb{N}$.)

3. In General if you want to show that two sets $\displaystyle A \text { and } B$ are equal show that

$\displaystyle A \subset B$ and $\displaystyle B \subset A$

Notice that for all $\displaystyle x \in \mathbb{N}$ that
$\displaystyle \displaystyle x \in A_x \implies x \in \cup_{n \in \mathbb{N}}^{\infty}A_n$

This should get you started

4. Got it. Thank you both for responding.