Find all values of $r \ge 0$ such that $x_n = \frac{r^n}{r^n + 1}$ is Cauchy.

Question

I figured that the best way to solve this would be to find all values of $r$ for which $x_n$ converges because, by definition, convergent sequences are Cauchy.

Is it sufficient to say that

• when $r \lt 1$, the sequence converges to $\frac{0}{0 + 1} = 0$
• when $r = 1$, the sequence converges to $\frac{1}{1 + \frac{1}{1^n}} = \frac12$
• when $r \gt 1$, the sequence converges to $\frac{1}{1 + \frac{1}{r^n}} = 1$

And thus, $\forall r \ge 0, x_n$ is Cauchy?

Answer 1

The poster of this problem has given a correct solution.So the answer to his question -"Is this correct " is Yes .