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Part CCSIR NET June 2023subsequence-criteria

Subsequence criteria

Under which of the following conditions is the sequence {xnx_{n}} of real numbers convergent?

  1. A.The subsequences {x2nx_{2n}₊1_{1}}, {x2nx_{2n}} and {x3nx_{3n}} are convergent and have the same limit.✓ Correct
  2. B.The subsequences {x2nx_{2n}₊1_{1}}, {x2nx_{2n}} and {x3nx_{3n}} are convergent.✓ Correct
  3. C.The subsequences {xknx_{kn}}n_{n} are convergent for every k ≥ 2.
  4. D.lim |xnx_{n}₊1−xn_{1} - x_{n}| = 0.

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: aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L

More on this topic

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

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

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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