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Cantor Intersection Theorem

  • Definition: Let {Q_1, Q_2, ... } be a countable collection of nonempty sets in R^n such that: i) Q_(k+1) is a subset of Q_k (where k=1, 2, 3, ...). ii) Each set Q_k is closed and Q_1 is bounded. Then the arbitrary intersection of Q_k as k ranges from 1 to infinity is closed and nonempty.

    Source: Mathematical Analysis, second edition by Tom M. Apostol

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