Is the Standard Proof of [a,b] in R Being Compact Flawed?

[ice109]

I have a bone to pick with the standard proof of the closed interval in R being compact with respect to the usual topology.

The proof starts out claiming that we can divide the interval in two and it is one of these two halves that is not coverable a finite subcollection of any open cover. Then we proceed to cut that interval in half and make the same claim and so on and so on.

The bone that I pick is what if both halves of the initial interval are uncoverable or both quarters of the initial half and so on.

 

[LCKurtz]

If both halves are uncoverable at any step, pick either one.

 

[ice109]

then why even have any discussion about "picking."?

 

[LCKurtz]

I don't know exactly what proof you are using but I'm guessing that you are showing you can construct a nested sequence of uncoverable intervals whose lengths go to zero, probably looking for a contradiction in an indirect argument. So you just have to say how you do each step. You always pick the "uncoverable" half or, if both halves are uncoverable, pick either one because it doesn't matter.