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Part CCSIR NET December 2024running-extremes-are-not-tail-extremes

Running extremes are not tail extremes

Let (an)n≥1,(bn)n≥1(a_{n})_{n}\ge_{1}, (b_{n})_{n}\ge_{1} and (cn)n≥1(c_{n})_{n}\ge_{1} be sequences given by an=(−1)n(1+ea_{n} = (-1)^{n}(1 + e⁻n),bn=max⁡^{n}), b_{n} = \max{a1,…,ana_{1}, \dots, a_{n}}, and cn=min⁡c_{n} = \min{a1,…,ana_{1}, \dots, a_{n}}. Which of the following statements are true?

  1. A.(an)n≥1(a_{n})_{n}\ge_{1} does not converge.✓ Correct
  2. B.lim⁡sup⁡(n→∞)an=lim⁡(n→∞)bn\lim \sup(n\to\infty) a_{n} = \lim(n\to\infty) b_{n}
  3. C.lim⁡inf⁡(n→∞)an=lim⁡(n→∞)cn\lim \inf(n\to\infty) a_{n} = \lim(n\to\infty) c_{n}
  4. D.lim⁡(n→∞)bn=lim⁡(n→∞)cn\lim(n\to\infty) b_{n} = \lim(n\to\infty) c_{n}

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: limsup(aₙ + bₙ) = limsup aₙ + limsup bₙ

More on this topic

The chapter behind this: limsup and liminf — three equivalent definitions and the algebra — free to read

From The Real Line › limsup, liminf and subsequential limits

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