Cauchy and monotone sequences

Question

I'm trying to come up with examples of sequence that relate to Cauchy and monotone in different ways. For example, I am trying to find a Cauchy sequence that is monotone and a monotone sequence that is not Cauchy. Help please!

Answer 1

Remember that every convergent sequence is Cauchy, so for "a Cauchy sequence that is monotone" any simple sequence like $(1/n)_{n \geq 1}$ will do.

For a monotone sequence that is not Cauchy, try $(n)_{n \geq 1}$. Try $\epsilon = 1/2$ for checking that it is not Cauchy.

Answer 2

The sequence $x_n = 2^{-n}$ is a monotone (decreasing) Cauchy sequence (it converges to $0$), but the sequence $y_n = n$ is a monotone (increasing) sequence that is not Cauchy (because it's unbounded).