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Let $\{x_n\}$ be a Cauchy sequence of nonnegative numbers. Prove that $$\{\sin (x_n + 5)^{1/3}\}$$ is a Cauchy sequence by checking the definition of Cauchy sequence.

I tried $$|x_m-x_n| < \epsilon$$ $$|\sin(x_m+5)^{1/3} - \sin(x_n+5)^{1/3}|$$ and apply sum to product. Please help for the following steps!!

naslundx
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SinSin
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1 Answers1

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Hint: $$a^3-b^3=(a-b)(a^2+ba+b^2),$$ and thus $$a-b=\frac{a^3-b^3}{a^2+ba+b^2}.$$

Set $a=\sin(x_m+5)^{1/3}$ and $b=\sin(x_n+5)^{1/3}$ and use the fact that $$\sin(x)-\sin(y)=2\cos\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right).$$

Surb
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