Problem: Let

be a metric space that satisfies the Bolzano-Weierstrass Property. Also, let

and

be disjoint, compact subsets of

.

is defined as

. Show that

.

Proof:

Suppose that

= 0. Then either 1)

or 2) Given any positive number, there exists a point in A and a point in B whose distance apart from eachother is less than this positive number. If it is 1), then

, contradicting the disjointedness of

and

.

So assume 2). We will construct two sequences. First take the number

. There exists points in

and

, let's call them

and

, such that

. Now consider the number

. There exists points in

and

, let's call them

and

, such that

. Now consider the number

. There are points in

and

, let's call them

and

, such that

. We can continue on like this and construct two sequences,

and

such that

.

By compactness of

, there is a subsequence

of

converging to

. Since

, then we see that the subsequence

also approaches

.

is closed, so this point

must also lie in

. Thus

, contradicting the disjointedness of

and

.

Thus

. QED.

Is this correct?