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I need to find the supremum and infimum of the set :

$S = \left[(-1)^n\left(4 - 1/n\right): n \in N\right]$

Calculating the few terms of sequence max S = $15/4$ ans min S = $-7/4$ Since both of these belong to the set they must be the supremum and infimum of the set.

Is my answer correct ?

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  • $\begingroup$ Why would you think that? $\endgroup$
    – lulu
    Sep 28, 2019 at 17:43

1 Answer 1

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I notice that $(-1)^{100}(4 - \frac{1}{100}) = \frac{399}{100}$ is larger than $\frac{15}{4}$, so $\frac{15}{4}$ cannot be the max.

Furthermore, I notice that $(-1)^{1000000}(4 - \frac{1}{1000000}) = \frac{3999999}{1000000}$ is larger than $\frac{399}{100}$, so even $\frac{399}{100}$ cannot be the max.

If you can begin to see a pattern here, then perhaps you can formulate the correct guess as to the supremum of $S$. And then you can apply a similar analysis using odd powers of $-1$ to correctly guess the infimum.

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  • $\begingroup$ Do you mean to say $4$ is a supremum of this sequence ? $\endgroup$
    – zeroflank
    Sep 28, 2019 at 18:02
  • $\begingroup$ Please can you tell me whether I am right or wrong. $\endgroup$
    – zeroflank
    Sep 28, 2019 at 18:18
  • $\begingroup$ Yup, that's it. $\endgroup$
    – Lee Mosher
    Sep 28, 2019 at 18:21

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