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

The supremum is approached and not attained

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

A.S is an unbounded subset of $R$ .
B.S is a bounded subset of $R$ and sup(S) = 1.✓
C.There exists u ∈ X such that ‖u‖ = 1 and T(u) = 1.
D.S is a closed subset of $R$ .

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.

See pricing

50 are analysed free — try those first.

Related counterexample: Every ordered field is Archimedean

The supremum is approached and not attained

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