Does the set $\lim_{n\to\infty}\{0/n,1/n,...,n/n\}$ contain irrational members? Does it equal the interval [0,1]?

Question

Please pardon my lack of mathematical sophistication, but I wonder whether such a set of rational elements would converge upon a closed interval as the divisor, and therefore the number of elements in the set, approaches infinity, and the distance between the elements becomes smaller and smaller.

Answer 1

You need to specify which limit you consider (there are several different definitions). But if you consider simply the set of limits of all converging sequences of rational numbers then the limit is the unit closed interval.You get the same result if you consider the Hausdorff limit.

Answer 2

Let's carefully see what you are asking for: Let $$S_1=\{0,1\},\quad S_2=\left\{0,\dfrac12, 1\right\},\quad S_3=\left\{0, \dfrac13, \dfrac23, 1\right\},\quad\cdots,$$ than you want to know if this sequence of sets converge to some other set. Before this we need to have a meaning (precise definition) for convergence of a sequence of finite sets inside some bigger space containing it. We can resolve this situation by introducing a appropriate (in general) topology, or a (complete) metric on the latter set. There are multiple ways to do this and I do not see any canonical way to do this.

Let me give you a toy example of the procedure I explain above. Consider the set $S$ of all subsets of $[0, 1]$ with discrete metric, that is,
$$d(S^1, S^2)= \begin{cases} 0, & \text{if $S^1=S^2$} \\ 1, & \text{otherwise} \end{cases}.$$

Here all Cauchy sequences sequences are eventually constant and hence, in this model, your sequence $\{S_1, S_2, S_3, \cdots\}$ does not converges.

Answer 3

The infinite union isn't a limit. It's an infinite union and it doesn't contain anything not in any of the sets.

So no, there union contains no irrational number.

But for any rational $r\in [0,1]$ we have $r = \frac ab, a,b \in \mathbb N$ and $a < b$ so $r=\frac ab \in \{0,\frac 1b, \frac 2b, ...., \frac bb\}$.

So the infinite union is $[0,1]\cap \mathbb Q = $ all the rational numbers in the interval $[0,1]$.

There's no such thing as sequence of sets converging to a set as a limit.