What is the diameter of the set $X_{n}=\{x_{n}, x_{n+1}, x_{n+2},...\}$ for the sequence $ x_{n} = \frac{(-1)^{n}}{n} $, with $n \in \mathbb{N}$

Question

I also don't know how to prove that sequence is Cauchy. Help please?

Answer 1

As $|x_k|<|x_{n+1}|<|x_n|$ for every $k>n+1$ and as $x_n$ and $x_{n+1}$ have opposite signs we have $$ \text{diam}(X_n):=\sup\{|x-y|:x,y\in X_n\}=|x_n-x_{n+1}|=\frac{1}{n}+\frac{1}{n+1}. $$ Now to prove $x_n$ is a Cauchy sequence just notice that $|x_n-x_m|\leq\text{diam}(X_n)$ for $m\geq n$.

Answer 2

$\{x_n\}$ is a Cauchy sequence if and only if $\lim_{N\to ∞}diam X_N = 0$.
The diameter of the set is the greatest distance two points in your set have, which in this case is $|x_n - x_{n+1}|$. Can you argue why?