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A monotone sequence that diverges but has a convergent subsequence.

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In Stephen Abbott textbook the solution states that

Impossible. This convergent subsequence would then be bounded; however, this would imply that the original sequence was also bounded. Because the original sequence is monotone, we know it cannot be bounded because we are told it diverges.

However for the sequence ${0,1,0,2,0,3,0,4,...}$ there is a subsequence which is bounded ${0,0,0,0,...}$ but the original sequence is unbounded. This is opposite to the solution which says "however, this would imply that the original sequence was also bounded".


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