question on "If a set is unbounded, then it cannot be compact" Hello, I am not a mathematician so wanted to understand by picturizing and got stuck in between. While trying to understand the proof as given in Wikipedia http://en.wikipedia.org/wiki/Heine-Borel_theorem I was not sure if I understood this statement very clearly. It says the following If a set is unbounded, then it cannot be compact Why? Because one can always come up with an infinite cover, whose elements have an upper finite bound to their size, i.e. the elements of the cover are not allowed to grow in size without bound. Please confirm my understanding here an example of an unbounded set is R then how is it that u can cover R with any (a,b) which are bounded ie how can u cover while |a| < some x and |b| < some y. The second problem refers to this particular statement I was trying to understand the theorem and so got confused with this line http://www.du.edu/~etuttle/math/heinebo.htm Now, the theorem says that in any such case, the closed interval a < x < b can be covered by a finite number of finite intervals. If we choose δ0 to be the smallest of the finite number of values of δ(ci) at the points ci, then we have that |f(x) - f(c)| < ε, for any ε > 0, whenever |x - c| < δ0, independently of c. So if I am understanding correctly it means there are |x1-c| < delta1 |f(x1) -f(c)| < epsilon1 |x2-c| < delta2 |f(x2) -f(c)| < epsilon2 surely |x1| < |x2| if delta1< delta2 and is epsilon1 <= epsilon2 ? If the above is true then how is |f(x) -f(c)| < any epsilon greater than 0. I can understand that it will be true for any epsion greater than equal to the smallest epsilon in that set but not sure how for all cases greater than 0 it is true.