Part BCSIR NET June 2024sup-need-not-be-attained

Sup need not be attained

Consider the set A = { $x \in Q : 0 < (2 - 1) x < 2 + 1$ } as a subset of $R$ . Which of the following statements is true?

A. $sup A = 2 + 23$
B. $sup A = 3 + 22$ ✓
C. $in f A = 2 + 23$
D. $in f A = 3 + 22$

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

50 are analysed free — try those first.

Related counterexample: Every ordered field is Archimedean

More on this topic

on a finite set every function is continuousDecember 2024
only an interval condition leaves uncountably manyDecember 2024
a connected domain forces a constant mapDecember 2024
the supremum is approached and not attainedDecember 2024
interval has cardinality of RJune 2024
cantor no set of that sizeJune 2024

The chapter behind this: The real line: supremum, Archimedes and density — free to read

From The Real Line › Completeness, sup/inf, Archimedean property

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