## 5.2. Compact and Perfect Sets

### Examples 5.2.2(a):

Is the interval [0,1] compact ? How about [0, 1) ?

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- The interval [0, 1] is compact. To see this, take any sequence of points in [0, 1].
Since the sequence must be bounded, we can extract a convergent subsequence by the
Bolzano Weierstrass theorem. Using the theorem on accumulation and boundary points
and noting that the set [0, 1] is closed, the limit of this subsequence must be contained in
[0, 1]. Hence, the set is compact by definition.
- The interval [0, 1) is not compact. Consider the sequence { 1 - 1/n }. Then that
sequence is contained in [0, 1), and converges to 1. Therefore, every subsequence of it
must also converge to 1, which is not part of the original set. Therefore, the set can not be
compact.