Why do I need boundedness for a a closed subset of $\mathbb{R}$ to have a maximum?

Question

Doesn't closedness suffice? If not, do you have a counterexample?

Answer 1

No, being closed isn't enough. $\mathbb N$ is a closed subset of $\mathbb R$ without a maximum.

Answer 2

Yes, there are counterexamples: $\mathbb{R}$ is closed.

Answer 3

No, take the whole real line for example. It is closed, but does it have a maximum?