Prove that $\bigcup_{n=1}^\infty[1, 1+1/n] = [1, 2]$.

Question

I understand why the union is $[1, 2]$, and I know I need to show each is a subset of the other. I'm just having trouble figuring out how to actually go about showing that.

Answer 1

The proof should be simple. For $n=1$ you have the set $[1,2]$. Now, for $n>1$ the set $[1,1+1/n]$ is a proper subset of $[1,2]$. Hence the union is just the biggest.

Answer 2

You can reason this inductively on the right endpoint of the interval, $a_n =1 + {1\over n}$. You want to show that $\{a_n\}_{n=1}^\infty$ is a strictly decreasing sequence; that is, you want to show that $$a_n > a_{n+1} \quad\implies\quad {1 + {1\over n}} > 1 + {1\over n+1} \quad(n\in\mathbb N).$$

This is trivially true by simple rearrangement. By cancelling the 1's on both sides, we get: $$\begin{align}{1\over n} > {1\over n+1} &\quad\implies\quad (n+1) > n \tag{cross-multiply}\\ & \quad\implies\quad1 > 0 \tag{subtract $n$ on both sides}.\end{align}$$

Obviously, $1$ is always greater than $0$, meaning this is a vacuous proof. Thus, the statement is true for any natural number $n$, which means that $$[1,a_1] \supset [1, a_2] \supset [1, a_3] \supset \cdots$$

Allowing you to say that $$\bigcup_{n=1}^\infty[1,a_n] = [1,2].$$