Proposition: Characterizing lim sup and lim inf

Let be an arbitrary sequence, and let c = lim sup aj and d = lim inf aj. Then
  1. there is a subsequence converging to c
  2. there is a subsequence converging to d
  3. d lim inf lim sup c for any subsequence {}
If c and d are both finite, then: given any > 0, there are arbitrary large j such that aj > c - and arbitrary large k such that ak < d + .

Proof:

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