Question

Prove {0,2,0,2,0,2,...} does not converge

Answer 1

Recall that a sequence $x_n$ is convergent if and only if $x_n$ is also a Cauchy sequence, which means to say that for any $\varepsilon>0$, we can find a sufficiently large $N$ for which

$|x_m-x_n|$

whenever both $m$ and $n$ exceed $N$.

But this never happens if we choose $m=n+1$ and $0$; under these conditions, we have

$|x_m-x_n|=|x_{n+1}-x_n|=2\not$

Therefore $x_n$ is not a Cauchy sequence and hence does not converge.