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Part CCSIR NET December 2024the-parts-need-not-have-densities

The parts need not have densities

For a positive integer n and a subset S of the set of positive integers, let S(n) denote the set {s ∈ S | s ≤ n}. Let X be a subset of the set of positive integers such that lim⁡(n→∞)\lim(n\to\infty) |X(n)|/n = 1. Assume that there exist pairwise disjoint subsets X1,X2,…,X8X_{1}, X_{2}, \dots, X_{8} of X such that ⋃(i=1..8)Xi=X(i=1..8) X_{i} = X. Which of the following statements are true?

  1. A.lim⁡(n→∞)\lim(n\to\infty) |Xi(n)X_{i}(n)|/n exists for all 1 ≤ i ≤ 8.
  2. B.lim⁡inf⁡(n→∞)\lim \inf(n\to\infty) |Xi(n)X_{i}(n)|/n ≥ 0 for all 1 ≤ i ≤ 8.✓ Correct
  3. C.lim⁡sup⁡(n→∞)\lim \sup(n\to\infty) |Xi(n)X_{i}(n)|/n ≥ 1/8 for some 1 ≤ i ≤ 8.✓ Correct
  4. D.lim⁡sup⁡(n→∞)\lim \sup(n\to\infty) |Xi(n)X_{i}(n)|/n < 1/8 for all 1 ≤ i ≤ 8.

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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