Divergent and Cauchy series in $\mathbb{R}$. What am I doing wrong?

Question

I was learning this week about complete spaces, $\mathbb{R}$ being one of them. According to what I understood this means every Cauchy series is convergent and vice-versa.

But look at this series: $a_{n} = \frac{sin(n)}{n}+n$.

It's a Cauchy series because the terms are getting closer together when $n$ grows. And it is divergent because $\lim_{n \to +\infty} a_{n} = +\infty$. What is it wrong in this explanation ? Thanks.

Answer 1

It's a Cauchy series [sic] because the terms are getting closer together when $n$ grows.

This is not the definition of a Cauchy sequence. You may want to check the definition: $(a_n)_n$ is said to be Cauchy if $$ \forall \varepsilon > 0\, \exists N_\varepsilon\, \forall n\geq N_\varepsilon, m \geq 0, \qquad \lvert a_{n+m}-a_n\rvert \leq \varepsilon. $$

Now, your sequence clearly does not satisfy that. The difference between even consecutive terms becomes arbitrarily close to 1... More specifically, you have $\lvert a_{n+m}-a_n\rvert \geq m - \frac{2}{n}$ for all $n,m$.