Mathematical Analysis Cauchy Sequence

Question

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.

Answer 1

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.