I am preparing for my Real Analysis test and I am trying to solve the following:
Prove that $\left((-1)^{n}\right)$ is not Cauchy directly by the definition
What have I tried so far? I started by negating the Cauchy sequence definition and got the following:
There exists $\epsilon_{0} > 0$ such that for every $n_{0} \in \mathbb{N}$ there exists $n,m \geq n_{0}$ such that $\left|x_{n}-x_{m}\right| \geq \varepsilon_{0}$
Having done that, I did the following:
Choose $\epsilon_{0} = 2$. Now, given $n_{0} \in \mathbb{N}$, choose n even such that $n > n_{0}$ and $m = n + 1 > n_{0} + 1 > n_{0}$
Now, we see that $\left|x_{n}-x_{m}\right| = \left|1-(-1)\right| = 2 = \epsilon_{0}$
Does it look good? Any help is appreciated!
Thanks in advance!