Example: Is the set {1, 2, 3} compact ? How about the set N of natural numbers ?

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• The set {1, 2, 3} is compact. Take any sequence with elements from the set {1, 2, 3}. This sequence is bounded, so we can extract a convergent subsequence from it. Since the subsequence converges, it forms a Cauchy sequence. Therefore, consecutive numbers in this subsequence must eventually be closer than, say, 1/2. But then the subsequence must eventually be constant, and that constant must be either 1, 2, or 3. Therefore the subsequence converges to an element of the original set. But then the set {1, 2, 3} is compact. Note that a similar argument applies to any set of finitely many numbers.
• The set of natural numbers N is not compact. The sequence { n } of natural numbers converges to infinity, and so does every subsequence. But infinity is not part of the natural numbers.

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