Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

Question

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?

I know that Cauchy condition means that for each $\varepsilon>0$ there exists $N$ such that $d(x_p,x_q)<\varepsilon$ whenever $p,q>N$.

Answer 1

Let $$ s_{n}:= \sum_{i=1}^{n} \frac{1}{i}. $$ Then $$ s_{n+1}-s_{n} = \frac{1}{n+1} \to 0 $$ as $n\to \infty$. However, this sequence is not Cauchy, as for any $\varepsilon > 0$, there is no $N$ such that $ |s_{m}-s_{n}| < \varepsilon $ for all $n,m\ge N$. In fact, $$ \lim_{m\to\infty} |s_{m}-s_{n}| = \infty, $$ for any $n$.

The "for all" quantifier on $m$ and $n$ is what confuses people about Cauchy sequences.

Answer 2

The standard example is the sequence $(s_n)$ of partial sums of the harmonic series. Formally, $$s_n=\sum_{k=1}^n \frac{1}{k}.$$

Note that $d(s_n,s_{n+1})=\frac{1}{n+1}$. It is clear that $d(s_n, s_{n+1})\to 0$ as $n\to\infty$.

But the sequence $(s_n)$ is not Cauchy. For given any $m$, we can find $n$ such that $d(s_m,s_n)$ is arbitrarily large. This is because the sequence $(s_n)$ diverges to infinity. We omit the proof, since you have likely already seen a proof that $\sum_{k=1}^\infty \frac{1}{k}$ diverges.

Answer 3

$$\textstyle 0,\,{1\over 2},\, 1,\, {2\over 3},\,{1\over 3},\,{0},\,{1\over 4},\,{2\over 4},\,{3\over 4},\,1,\,{4\over 5},\,{3\over 5},\,{2\over 5},\,{1\over 5}, \,0, \ldots$$