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Counterexample

Is this true?

an1/na_{n}^{1/n} →L⇒an\to L \Rightarrow a_{n}₊1/an→L_{1}/a_{n} \to L

No — it is false.

The counterexample

The object that kills this claim comes with Trap Analysis.

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86 counterexamples are free — including most of the ones Part C leans on hardest.

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Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Go deeper· Explained in full in Sequences in ℝ — the implication map· Asked in CSIR NET December 2025· From The Real Line › Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

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