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Let $a_1,a_2,a_3,...$ and $b_1,b_2,b_3,...$ be Cauchy sequences in $[0,\infty)$, and let $c_n = a_n^2+\sqrt{b_n} + \sin(a_n+b_n)$. Prove that $c_1,c_2,c_3,...$ is also a Cauchy sequence by using the fact that a sequence of real numbers is a Cauchy sequence if and only if it converges.

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closed as off-topic by Delta-u, user370967, Martin Sleziak, Strants, Ethan Bolker May 24 '18 at 14:48

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  • $\begingroup$ Welcome to stackexchange. I'm voting to close this question because you show no work of your own. Please edit it to tell us what you tried and where you are stuck. Then perhaps we can help. $\endgroup$ – Ethan Bolker May 24 '18 at 14:48
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Since the sequences $(a_n),(b_n)$ are Cauchy sequences, they converge.

Suppose $(a_n)$ converges to $A$, and $(b_n)$ converges to $B$.

Since $b_n \ge 0$ for all $n$, it follows that $B \ge 0$.

Since the functions $x\mapsto x^2$ and $x\mapsto \sin(x)$ are continuous, and the function $x\to\sqrt{x}$ is continuous on the interval $[0,\infty)$, it follows that

  • $(a_n^2)$ converges to $A^2$.$\\[4pt]$
  • $(\sqrt{b_n})$ converges to $\sqrt{B}$.$\\[4pt]$
  • $(a_n+b_n)$ converges to $A+B$, hence $\bigl(\sin(a_n+b_n)\bigr)$ converges to $\sin(A+B)$

It follows that $(c_n)$ converges to $A^2+\sqrt{B}+\sin(A+B)$.

Hence, since the sequence $(c_n)$ converges, it's a Cauchy sequence.

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