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1. In the partially ordered set N × N with the ordering ( j, k) ≤ (m, n) iff j ≤ m and k ≤ n, consider the sequence (n, n), for n = 0, 1, 2,... . Describe all the maximal chains that include the given sequence. Do any of these chains have upper bounds?

2. Find a partially ordered set (X, ≤), with as few elements as possible, for which the hypothesis of Zorn's lemma holds (every chain has an upper bound) but which does not have a maximum.

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