Let $q_n, n=1,2,3,\ldots$ be on stringing of rational numbers from $[0,1]$. $a.) $Let $q_n, n=1,2,3,\ldots$ be on stringing of rational numbers from $[0,1]$ and let the be given a sequence of sets: $A_n=[q_n,1] n=1,2,3,\ldots$ Find $$\overline{\lim_{n\to \infty}} A_n.$$ Now this $n\to \infty$ isn't quite given but I can only pressume that I must be the case so I feel I should do the assignment like , because similar has been done in class: $$\overline{\lim_{n\to \infty}} A_n= \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k.$$ Can someone formally type out the answer, if possible using this method? $b.)$ And basically the same question: Let $q_n,\ n=1,2,3,\ldots$ be on stringing of rational numbers from $[0,1]$ and let the be given a sequence of sets: $A_n=[0,q_n],\ n=1,2,3,\ldots$ Find $$\underline{\lim} A_n.$$ Here in what I think are similar examples we did: $$\bigcup_{n=1}^\infty \bigcap_{k=n}^\infty A_k.$$

Hint for the first problem: Let $n$ be given, and let $\epsilon\gt 0$. Then there is a $k\gt n$ such that $q_k\lt \epsilon$. What does that say about $\bigcup_{k=n}^\infty A_k$?

**Added:** We first show that for all $n$ there is a $k\gt n$ such that $q_k\lt \epsilon$. This is because there are infinitely many rationals in the interval $0\lt x\lt \epsilon$, and there are only $n$ rationals of the form $q_i$ where $i\le n$.

So for every $\epsilon\gt 0$, there is a $k\gt n$ such that $[\epsilon,1]\subset A_k$. It follows that $\bigcup_{k=n}^\infty A_k$ contains the interval $(0,1]$. For every large enough $n$, $\bigcup_{k=n}^\infty A_k$ does not contain $0$. We conclude that $\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k=(0,1]$.

Hint for the second problem: Use the fact that for every $n$, and for every $\epsilon \gt 0$, there is a $k\gt m$ such that $q_k\gt 1-\epsilon$.