#### 4.7.1 Sequential Compactness

First I will discuss the very important concept of sequential compactness. This is a property
that some sets have. A set of numbers is sequentially compact if every sequence contained in
the set has a subsequence which converges to a point in the set. It is unbelievably useful
whenever you try to understand existence theorems.

Definition 4.7.1 A set, K ⊆ F is sequentially compact if whenever

⊆ K
is a sequence, there exists a subsequence, such that this subsequence converges to
a point of K.
The following theorem is part of the Heine Borel theorem.

Theorem 4.7.2 Every closed interval

is sequentially compact.
Proof: Let

⊆ ≡ I_{0}. Consider the two intervals

and

each of
which has length

∕2

. At least one of these intervals contains

x_{n} for infinitely many
values of

n. Call this interval

I_{1}. Now do for

I_{1} what was done for

I_{0}. Split it in
half and let

I_{2} be the interval which contains

x_{n} for infinitely many values of

n.
Continue this way obtaining a sequence of nested intervals

I_{0} ⊇ I_{1} ⊇ I_{2} ⊇ I_{3}
where the length of

I_{n} is

∕2

^{n}. Now pick

n_{1} such that

x_{n1} ∈ I_{1},

n_{2} such that

n_{2} > n_{1} and

x_{n2} ∈ I_{2},n_{3} such that

n_{3} > n_{2} and

x_{n3} ∈ I_{3}, etc. (This can be done
because in each case the intervals contained

x_{n} for infinitely many values of

n.) By
the nested interval lemma there exists a point

c contained in all these intervals.
Furthermore,

and so lim_{k→∞}x_{nk} = c ∈

. ■