The two definitions of a compact set 
*
* In general, $A$ is compact if every open cover of $A$ contains a finite subcover of $A$.

* In $R$, $A$ is compact if it is closed and bounded.

The second is very easy to understand because I can easily come up with an example like $[0,1]$ which is both closed and bounded so it's compact.
However, I am very confused at definition (1) because I don't really understand what is meant by a cover and I don't understand how this is really related to a set being closed and bounded?
Could someone please explain what is the relationship between (1) and (2)?
Thank you.

The relationship between 1 and 2 is due to the fact that R has the least upper bound property (aka R is complete). The proof of the equivalency can be found pretty easily online. Baby Rudin contains the proof in the 2nd chapter when Cauchy sequences are introduced.
