Can a sequence converge but be not Cauchy? Any examples?

Question

I know that not every Cauchy sequence is convergent. How about the other way around? Can someone bring up an example?

Thanks in advance!

Answer 1

Hint:

$$|x_n-x_m| = |x_n - L + L - x_m| \leq |x_n-L|+|x_m-L|.$$

The analogous thing works in metric spaces. This trick is very common in lots of situations in analysis, so it would do you well to understand it.

Answer 2

But Note that in general Converse is not true i.e. A Cauchy sequence is not necessarily a convergent sequence. For example if our space is $X=\mathbb Q$, then $$ x_n=\frac{\lfloor n\sqrt{2}\rfloor}{n}, $$
is a Cauchy sequence which DOES NOT converge is $\mathbb Q$. It DOES converge is $\mathbb R$ but not in $\mathbb Q$.