Give an example of a set S and a sequence (xn) that is frequently in S and a partial limit of x o...

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Give an example of a set S and a sequence (xn) that is frequently in S and a partial limit of x o...

Give an example of a set S and a sequence (x_n) that is frequently in S and a partial limit of x of (x_n) such that x fails to be close to S. (Real Analysis)

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Answer 1

Answer #1

$S=[0,1]$ and the sequence is $rn = (-1)n$

Thus, $x_n=1\in S$ for even n which happens infinitely often

But $x_{n}=-1$ for odd n which is not in S

So the partial limit $-1$ fails to be close to S

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Answer 2

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