Real numbers/Bounded, not convergent sequence/pm 1/Example

The alternating sequence

${}x_{n}:=(-1)^{n}\,$

is bounded, but not convergent. The boundedness follows directly from ${}x_{n}\in [-1,1]$ for all ${}n$. However, there is no convergence. For if ${}x\geq 0$ were the limit, then for positive ${}\epsilon <1$ and every odd ${}n$ the relation

${}\vert {x_{n}-x}\vert =\vert {-1-x}\vert =1+x\geq 1>\epsilon \,$

holds, so these members are outside of this ${}\epsilon$-neighbourhood. In the same way we can argue against some negative limit.