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Part CCSIR NET June 2023subsequence-criteria

Subsequence criteria

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

  1. A.The subsequences {₊}, {} and {} are convergent and have the same limit.✓
  2. B.The subsequences {₊}, {} and {} are convergent.✓
  3. C.The subsequences {} are convergent for every k ≥ 2.
  4. D.lim |₊| = 0.

Solution

(1) Even and odd subsequences with a common limit ⇒ convergent. (2) {} lies in both {} and {}, so those limits agree; {₊} lies in both {₊} and {}, so all three limits agree ⇒ convergent. (3) Fails: if n is prime, 0 otherwise — every {} is eventually 0 but diverges. (4) Fails: .

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

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

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