Estimate for $|x_n-A|$ for a Cauchy sequence $(x_n)$ satisfying $|x_{n+1}-x_n|<Cr^n$ for some constant $C$, and $\lim_{n \rightarrow \infty} x_n=A$.

Question

Find an estimate for $|x_n-A|$ in terms of $n$ and $r$, where $(x_n)$ is a Cauchy sequence satisfying $$|x_{n+1}-x_n|<Cr^n$$ for some constants $C$, $0<r<1$, and $\lim_{n \rightarrow \infty} x_n=A$.

How can I use the concept of Cauchy and its convergence to find the estimate?

Answer 1

Knowing that

$$|x_{n+1}-x_n|<Cr^n$$ for all $n \in \mathbb N$, you get for $m \gt n \in \mathbb N$

$$\begin{aligned}\vert x_m - x_n \vert &\le \vert x_m - x_{m-1} \vert + \dots + \vert x_{n+1}-x_n \vert\\ &\le Cr^{m-1} + \dots + Cr^n\\ &\le Cr^n + C r^{n+1} + \dots\\ &= \frac{Cr^n }{1-r} \end{aligned}$$

Now, in this inequality that is valid for all $m \gt n$ make $m \to \infty$. You get

$$\vert x_n - A \vert \le \frac{Cr^n }{1-r}.$$