Monotone sequences

  • Thread starter mathanon
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mathanon
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Let A be a nonempty subset of R that is bounded above and let α=supA. Show that there exists a monotone increasing sequence {an} in A such that α=lim an. Can the sequence {an} be chosen to be strictly increasing?
 

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HallsofIvy
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For every positive n, look at the set [itex]\{ a_i| |a_i- a|<1/n\}[/itex]. Can you see that this set is non-empty for all n? Choose a member of this set to be in the subsequence.
 

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