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What is the supremum of the set consisting the following real numbers:

• 0.200 . . .

• 0.2500 . . .

• 0.25200 . . .

• 0.252500 . . .

Would it just be 0.252500?

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  • $\begingroup$ Just these four numbers? Then, yes. The supremum of a finite set is the largest element of that set. But it looks like you are just writing the first terms of a sequence. Could you clarify? $\endgroup$
    – lulu
    Nov 19, 2019 at 19:18
  • $\begingroup$ Assuming the pattern continues the way I think it does, $0.2525$ is going to be too small to be a supremum, as the next number in the sequence, $0.25252$ is larger. $\endgroup$
    – Arthur
    Nov 19, 2019 at 19:19

1 Answer 1

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The supremum is

$$25 \left(\frac{1}{100} + \left(\frac{1}{100}\right)^2 + \dots\right) = \frac{25}{100}\frac{1}{1-\frac{1}{100}}=\frac{25}{99}$$

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  • $\begingroup$ Which is indeed 0.2525252525.... (25 being repeated indefinitely) $\endgroup$
    – Jeanba
    Nov 20, 2019 at 6:44

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