TITLE: What is the supremum of the set?
QUESTION [3 upvotes]: 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?

REPLY [2 votes]: 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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