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Part CCSIR NET December 2025split-into-odd-and-even-n-separately-one-half-stays-bounded-the-other-diverges

Split into odd and even n separately one half stays bounded the other diverges

Consider the following subset of real numbers A = {(1+(−1)n)n−1/n:n(1+(-1)^{n})n - 1/n : n is a positive integer}. Which of the following statements are true?

  1. A.A is bounded below but not bounded above.✓ Correct
  2. B.A is bounded above but not bounded below.
  3. C.inf A = −1✓ Correct
  4. D.sup A = 1

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 limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

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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