Is every Cauchy sequence contractive?

Question

I know that a contractive sequence is a Cauchy sequence. Is the converse true? Does a sequence being Cauchy implies that it is contractive, if the sequence is defined over the set of real numbers? If not, please provide a mathematical explanation/proof supported with a counter-example.

Answer 1

No. For example consider $a_n = 1/n$.

Then $\frac{|a_{n+2}-a_{n+1}|}{|a_{n+1}-a_n|} = \frac{n(n+1)}{(n+1)(n+2)} \to 1$ as $n \to \infty$.

Answer 2

I assume that the underlying metric space is $\mathbb{R}$.

Since every Cauchy sequence is convergent and since there exist non-contractive convergent sequences, I guess that the answer is "No".