[Math] Does $\bigcap_{n=1}^{+\infty}(-\frac{1}{n},\frac{1}{n}) = \varnothing$


When I learn the below theorem:

If $I_n$ is closed interval, and $I_{n+1} \subset I_n$, then $$\bigcap I_n \ne \varnothing$$

and someone says if we replace closed interval with open interval, can construct counter-example.

So I have tried to construct the one: Does $$\bigcap_{n=1}^{+\infty}\left(-\frac{1}{n},\frac{1}{n}\right) = \varnothing\quad?$$

Thanks very much.

Best Answer

No. It is not empty. Since $0\in (-1/n, 1/n)$ for all $n$, so $0\in\bigcap_{n=1}^\infty \left(-\frac{1}{n},\frac{1}{n}\right)$.

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