15.4 Sequential Compactness
The concept of sequential compactness is also the same as before.
Definition 15.4.1 A set K in ℝp is sequentially compact if every sequence
in K has a subsequence which converges to a point in K.
Theorem 15.4.2 Let K be a nonempty subset of ℝp. Then K is
sequentially compact if and only if it is closed and bounded.
Proof: Suppose first that K is closed and bounded. Then by definition,
for a suitable product of closed and bounded intervals. Let the
Then it follows from the definition of the
Cartesian product that for each i,xik ∈
convergent subsequence, denoted by x1k1
such that limk
a convergent subsequence denoted as
converging to x2 ∈
that if a sequence of real numbers converges, then so does every subsequence.
It follows that lim
. Continue taking subsequences such that
= xj ∈
j ≤ r
. Therefore, limk
for each i ≤ p
and this shows that limkp→∞xkp
is closed and so x ∈ K
. This shows that a closed and bounded nonempty set is
Conversely, suppose a set K is sequentially compact. Then the set must be bounded
since otherwise one could obtain a sequence of points of K which is unbounded. This
sequence cannot be a Cauchy sequence and so it cannot converge. If the set is not closed,
then by Theorem 15.2.4 above, there would be a point x
and a sequence of
points of K
which converges to x
. But now this sequence must have a
convergent subsequence converging to a point of K
. This is impossible because all
subsequences must converge to x
which is not in K
. Therefore, K
must also be closed.