Finding and Proving the Supremum and Infimum of $X = \bigcup\limits_{n=1}^{\infty} [2n, 2n + 1]$

Question

$X = \bigcup\limits_{n=1}^{\infty} [2n, 2n + 1]$

So proofing these types questions involve usage of the Archemedian Property and some inequalities from what I've found. But with a union of infinite sets I'm sort of lost. It seems trivially the Infimum of X will be 2. And Supremum I have nothing except intuition it doesn't have a maximum/supremum.

How would you proceed with this type of problem? Construct an inequality and use the Arch. Property to show a greatest lower bound?

Thank you in advance. Have a good day!

Answer 1

$\inf(X) = 2$, already dealt with.

Show that $X$ is not bounded above.

Assume $B \gt 0$ is an upper bound.

Archimedes:

There is a $n_0$ such that $n_0 > B/2.$

Hence for $n \ge n_0:$

$B \lt 2n_0 \le 2n $.

Consider $x \in [2n,2n+1] $ , then $x \in X$, and $B< x.$

$B$ is not an upper bound of $X$.

$\rightarrow:$

$\sup(X) = \infty.$

Answer 2

Clearly $$\inf X = \min X =2$$ and if we suppose $s= \sup X<\infty$ then there exist $n\in \mathbb{N}$ such that $s\in[2n,2n+1]$. But then $s<2n+2\in X$ a contradiction. So $s= \infty$