Every convergent sequence is a Cauchy sequence.

Question

Today, my teacher proved to our class that every convergent sequence is a Cauchy sequence and said that the opposite is not true, i.e. Not every Cauchy sequence is a convergent sequence. However he didn't prove the second statement. Is there an example or a proof where Cauchy sequence is not convergent?

Answer 1

In the metric space $(0, 1]$, the sequence $(a_n)_{n=1}^\infty$ given by $a_n = \frac{1}{n}$ is Cauchy but not convergent.

Answer 2

You will not find any real-valued sequence (in the sense of sequences defined on $\mathbb{R}$ with the usual norm), as this is a complete space. (By definition, a metric space is complete if... every Cauchy sequence in this space is convergent.)

But you can find counter-examples in more "exotic" metric spaces: see, for instance, the corresponding section of the Wikipedia article. One of the classical examples is the sequence (in the field of rationals, $\mathbb{Q}$), defined by $x_0=2$ and $$ x_{n+1} = \frac{x_n}{2} + \frac{1}{x_n} $$ for $n \geq 0$. It can be shown this sequence is Cauchy; but it converges to $\sqrt{2}$, which is not a rational: so the sequence $(x_n)_{n\geq 0}$ is Cauchy (in $\mathbb{Q}$), but not convergent (in $\mathbb{Q}$).

(Note that the same sequence, if defined as a sequence in $\mathbb{R}$, does converge, as $\sqrt{2}\in\mathbb{R}$).