Is the sequence $(1 + (-1)^n)$ Cauchy? We're supposed to prove or disprove the statement that $(1 + (-1)^n)$ is a Cauchy sequence. I don't think it is because this function oscillates and doesn't converge, so I don't think it's Cauchy. Why or why not is this a Cauchy sequence? Thanks.
 A: Note that $\left \vert a_{n+1} - a_n \right \vert = 2$ for all $n$. So is this sequence Cauchy?
A: You can apply the definition of Cauchy sequence directly. Note that if $a_n=1+(-1)^n$, then
$$a_n=\begin{cases}
2,&\text{if }n\text{ is even}\\
0,&\text{if }n\text{ is odd}\;.
\end{cases}$$
Let $\epsilon>0$. If the sequence is Cauchy, there must be an $m\in\Bbb N$ such that $|a_k-a_\ell|<\epsilon$ whenever $k,\ell\ge m$. Now consider $|a_k-a_{k+1}|$ for any $k\in\Bbb N$; it’s equal to ... ?
A: If the series converges to a limit $L$, then for sufficiently big values of $n$, you'd have $(1+(-1))^n$ between $L\pm1/10$.  Thus two consecutive terms would differ from each other by less than $2/10$.  But $0$ and $2$ do not differ by less than $2/10$.  Hence the sequence diverges.  Therefore it's not a Cauchy sequence.
PS: There are some who say that although some sequences are Cauchy sequences, no sequence is Cauchy, since Cauchy is a man who lived in the 19th century.  The idea is that the term "Cauchy sequence" is a compound noun, rather than "Cauchy" being an adjective.  I sympathize with this view, but mathematicians generally don't even suspect there is such a view.  Language is messy.
