Let $(a_n)_n$ be a Cauchy sequence of rational numbers. Prove that $\dfrac{4a_n^4}{2 + a_n^3}$ is also a Cauchy sequence.
I know that Cauchy sequences can be added, subtracted and multiplied but I'm not sure how to apply this to the proof.
2026-03-25 01:16:42.1774401402
Cauchy Sequence of Rational Numbers - Prove another sequence is Cauchy
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You can't. Let $(a_n)_n$ be a sequence of rational numbers converging to $-\sqrt[3]{2}$. (The choice of rational $a_n$ is possible thanks to the density of $\Bbb Q$ in $\Bbb R$.) Since it's convergent, it's Cauchy, and thus bounded.
The fraction $\dfrac{4a_n^4}{2+a_n^3}$ has bounded nominator, but its denominator tends to zero as $n$ tends to infinity. Therefore, this fraction is not bounded, thus it's not Cauchy.