Why is any unbounded subset of the reals not compact?

Question

Why is any unbounded subset of the reals not compact?

I have that a subset of the reals is compact if every open cover of this subset has a finite sub-cover.

Answer 1

For your unbounded subset of the reals, consider the open cover that consists of the sets $\{O_1, O_2, \dotsc\}$ where $O_i$ is the open ball centered at the origin with radius $i$. This cover admits no finite subcover.

Answer 2

Take the cover $\{(-n,n)\}_{n\in\Bbb N}$.

Answer 3

An alternative definition for compactness, which is logically equivalent to the other one says that a subset of $\mathbb R$ is compact if and only if it is closed and bounded.

Some analysis textbooks take this as THE definition, other use it as an if and only if theorem.

Now you know. It can't be compact because it's not bounded.