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Provide an example (and prove that it works) or prove that there’s no such example exists. a) A bounded sequence that does not converge to 4/9 but has a subsequence converging to 4/9

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Provide an example (and prove that it works) or prove that there’s no such example exists.

a) A bounded sequence that does not converge to 4/9 but has a subsequence converging to 4/9

b) A monotone sequence that does not converge to 4/9 but has a subsequence converging to 4/9

c) A sequence with both an increasing subsequence and a decreasing subsequence that does not converge.

 
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