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Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

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#1 · Analysis & Linear Algebra › The Real Line

“Cesàro means converge ⇒ the sequence converges” — false

Counterexample: aₙ = (−1)ⁿ

Partial averages → 0, sequence diverges.

sequences

#2 · Analysis & Linear Algebra › The Real Line

“aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L” — false

Counterexample locked — unlock with Notes + PYQ

sequences

#3 · Analysis & Linear Algebra › The Real Line

“limsup(aₙ + bₙ) = limsup aₙ + limsup bₙ” — false

Counterexample: aₙ = (−1)ⁿ, bₙ = (−1)ⁿ⁺¹

$a_{n} + b_{n} \equiv 0$ , so the left side is 0 while the right side is 1 + 1 = 2.

sequenceslimsup