Part CCSIR NET December 2024rationalise-and-the-sequence-becomes-monotone

Rationalise and the sequence becomes monotone

Let A = { $1/ (n + n - n)$ | $n \in Z, n > 0$ }. Which of the following statements are true?

A.sup A is finite.✓
B. $lim sup (n \to \infty) 1/ (n + n - n) < sup A$ ✓
C. $lim in f (n \to \infty) 1/ (n + n - n) = 2$ ✓
D. $lim sup (n \to \infty) 1/ (n + n - n) = 3$

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.

50 are analysed free — try those first.

Related counterexample: aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L

More on this topic

subsequence criteriaJune 2023
recursive sequence fixed pointJune 2023
unbounded vs bounded sequencesDecember 2023
Part B questionDecember 2023
the two sequences differ by a null termDecember 2024
a ratio of growing terms can be boundedDecember 2024

The chapter behind this: Sequences in ℝ — the implication map — free to read

From The Real Line › Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

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