Question

4.4.3. Proposition. Every sequence of real numbers has a monotone subsequence.

strictly

Proof. Exercise. Hint. A definition may be helpful. Say that a term am of a sequence in R is a peak term if it is greater than or equal to every succeeding term (that is, if am ≥ am+k for all k ∈ N). There are only two possibilities: either there is a subsequence of the sequence (an) consisting of peak terms, or else there is a last peak term in the sequence. (Include in this second

Answer 1

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