planet gareth : maths : heine-borel theorem

A proof of the Heine-Borel Theorem, written by Ben Millwood.

This theorem's more than just a little fun:
let X be all the reals from nought to one
and give me S a set of open sets
together covering the whole of X.
All X can be enclosed by a subset
of S of finite size, and so we get
that X is a compact subset of R
and then so all closed bounded subsets are.

Now take a set from X and call it Y
containing all the x which satisfy
"from nought to x in n subsets from S
(some n which is of finite size, no less!)"
Since S covers all X and must contain
a single set which reaches nought, it's plain
that Y has elements, and since it's found
inside of X, it has an upper bound.

So sup of Y exists, let's call it t
and t's in Y: assume the contrary;
but then there's finite subsets up to t
and S covers all X, as previously,
so just one more subset will get t too
and n+1 subsets reach t - untrue!
Now t is 1: the open set it's in
contains a neighbourhood of t within,
and so if t (sup Y) was less than 1,
it covers more than that, and so we're done.
So t is 1, and shown to be in Y:
then Y is X. And so, easy as π,
we've proved that which we first set out to do:
the theorem of Heine-Borel is true!