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

Sup need not be attained

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

  1. A.sup⁡A=2+23\sup A = 2 + 2\sqrt{3}
  2. B.sup⁡A=3+22\sup A = 3 + 2\sqrt{2}✓ Correct
  3. C.inf⁡A=2+23\inf A = 2 + 2\sqrt{3}
  4. D.inf⁡A=3+22\inf A = 3 + 2\sqrt{2}

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, and why this one tests boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: Every ordered field is Archimedean

More on this topic

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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