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

Running extremes are not tail extremes

Let $(a_{n})_{n} \geq_{1}, (b_{n})_{n} \geq_{1}$ and $(c_{n})_{n} \geq_{1}$ be sequences given by $a_{n} = (- 1)^{n} (1 + e$ ⁻ $^{n}), b_{n} = max$ { $a_{1}, \dots, a_{n}$ }, and $c_{n} = min$ { $a_{1}, \dots, a_{n}$ }. Which of the following statements are true?

A. $(a_{n})_{n} \geq_{1}$ does not converge.✓
B. $lim sup (n \to \infty) a_{n} = lim (n \to \infty) b_{n}$
C. $lim in f (n \to \infty) a_{n} = lim (n \to \infty) c_{n}$
D. $lim (n \to \infty) b_{n} = lim (n \to \infty) c_{n}$

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.

See pricing

50 are analysed free — try those first.

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

Running extremes are not tail extremes

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