Recently, studying for a Master's admission test, I came across this question from a previous test.
Let $(x_n)$ be a sequence of real numbers with the following property: For every $n$ natural
$$\mid x_{n + 1} - x_n \mid \ < \ \dfrac{1}{n}.$$
$(x_n)$ converges or diverges?
I believe that information is missing, for example, the terms of the sequence are positive or negative or something like that. I have tried to prove that this sequence is cauchy but have not been successful. Here's my attempt:
Given $\varepsilon > 0$ arbitrary, exists $n_0 \in \mathbb{N}$ such that $\dfrac{1}{n_0} \ < \ \varepsilon$ by archimedean property. Let $m, n > n_0$ arbitrary. WLOG, suppose m $\geq$ n. We have that
$$\mid x_m - x_n \mid \ = \ \mid (x_m - x_{m -1}) + (x_{m - 1} - x_{m -2}) \ + \ ... \ + \ (x_{n + 1} - x_n)\mid $$
$$\Rightarrow \mid x_m - x_n \mid \ \leq \ \mid x_m - x_{m - 1} \mid \ + \ \mid x_{m-1} - x_{m -2} \mid \ + \ ... \ + \ \mid x_{n + 1} - x_n \mid$$
$$\Rightarrow \mid x_m - x_n \mid \ \leq \ \dfrac{1}{m - 1} \ + \ \dfrac{1}{m - 2} \ + \ ... \ + \ \dfrac{1}{n}. $$
But this does not provide us with any concrete information. Do you suggest something?
I have also tried to prove that a sequence satisfying this is either constant or of the form $\dfrac{1}{n^k}$, but without success.