Questions about infinite intersections of sets I am making some examples to make sure I understand these ideas correctly, but some of them im unsure of, I posted what I think the intersection are. Are they correct?


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*$\displaystyle\bigcap^{\infty}_{k=1}\left[1,1+\frac{1}{k}\right]=\{1\}.$

*$\displaystyle\bigcap^{\infty}_{k=1}\left(1,1+\frac{1}
{k}\right)=\emptyset.$

*$\displaystyle\bigcap^{\infty}_{k=1}\left(1,1+\frac{1}{k}\right]=\emptyset.$

*$\displaystyle\bigcap^{\infty}_{k=1}\left[1,1+\frac{1}{k}\right)=\{1\}.$

All four are correct. They are not justified, but they are correct
An example of a justification (for 3.):

Let $\displaystyle A = \bigcap_{k=1}^\infty(1, 1+\frac1k)$, and let $x\in\mathbb R$.
If $x\leq 1$, then obviously, $x\notin A$.
If $x>1$, then $x=1+\epsilon$ for some $\epsilon>0$. There then exists some $k$ for which $\frac 1k < \epsilon$, meaning that $x > 1+\frac1k$ and so $x\notin (1, 1+\frac1k)$. This also means that $x\notin A\subseteq (1, 1+\frac1k)$.

