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Part CCSIR NET December 2024the-supremum-is-approached-and-not-attained

The supremum is approached and not attained

Consider X = {u | u:[0,1]→Ru : [0, 1] \to \mathbb{R} is continuous and u(0) = 0} with the sup norm ‖u‖ = sup(x∈[0,1])|u(x)|. Let T(u)=∫01u(t)T(u) = \int_{0}^{1}u(t)dt and S = {|T(u)| : u ∈ X, ‖u‖ ≤ 1}. Which of the following statements are true?

  1. A.S is an unbounded subset of R\mathbb{R}.
  2. B.S is a bounded subset of R\mathbb{R} and sup(S) = 1.✓ Correct
  3. C.There exists u ∈ X such that ‖u‖ = 1 and T(u) = 1.
  4. D.S is a closed subset of R\mathbb{R}.

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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