Part CCSIR NET December 2023unbounded-vs-bounded-sequences

Unbounded vs bounded sequences

Let x be a real number. Which of the following statements are true?

A.There exists an integer n ≥ 1 such that $n^{2} sin (1/ n) \geq x$ .✓
B.There exists an integer n ≥ 1 such that n cos(1/n) ≥ x.✓
C.There exists an integer n ≥ 1 such that n $e^{- n}$ ≥ x.
D.There exists an integer n ≥ 2 such that n (log $n)^{- 1}$ ≥ x.✓

Solution

$n^{2} sin (1/ n)$ ~ n, n cos(1/n) ~ n and n/log n all tend to $+ \infty$ , so each eventually exceeds any x. n $e^{- n}$ ≤ 1/e is bounded, so it fails for x > 1/e.

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: aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L

Unbounded vs bounded sequences

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