Prove That a sequence of $\mathbb{Q}$ converges to an irrational number Let $A$ be the set $A = [0,1]\cap \mathbb{Q}$. My goal is to prove that A is not closed in $\mathbb{R}$.
I know there are many ways but in order to get good and understand a specific method I was asked to be more specific about my solution.
So my answer was that let a sequence $a_n,a_n\in[0,1]\cap \mathbb{Q}$.
And I supposed that $a_n$ converges to $1/\sqrt{2}$ which obviously is not in $[0,1]\cap \mathbb{Q}$ and therefore $A$ was not closed.
So I would like to kindly ask how should I be more specific about this but let me first tell you my thoughts. Let's forget what have been written until now and lets start over again.
$1)$ Let $x\in\mathbb{R}$ from the density of $\mathbb{Q}$, for every $n\in\mathbb{N}$ there is is a $q\in\mathbb{Q}$ between $x-1/n,x$ so that $x-1/n< q < x$
So for every $n\in\mathbb{N}$ I have that $A_n = (x - 1/n,x)\cap\mathbb{Q}\neq \emptyset$
$i)$ $q_n $ sequence and $q_n\in A_n $ plus $ q_n\in \mathbb{Q}$ for every $n\in\mathbb{N}$
$ii)$ also $x-1/n< q_n < x$ for every $n\in\mathbb{N}$ and since $x-1/n \to x$ and $ x\to x => q_n \to x$
So I actually proved that I can take a sequence in ${Q}$ that converges to a $x\in\mathbb{R}$.
Could you help me with the last part actually proving that $x \in \mathbb{R-Q}$ in my first example?
 A: Your proof is confusing and extremely hard to read. Here are a couple of suggestions.
First:
You define $A_n=\left(x-\frac{1}{n}, x\right)\cup \mathbb Q$... are you sure you want the union here? Did you mean intersection?
Second:
Out of nowhere, you start talking about the sequence $q_n$. Where did you define the sequence? You should be explicit about each step you are doing. So explain when a given equation is a definition, and when it is derived (and in that case, make it clear where the equation is derived from).
Third:
Improve your formatting. This includes:

*

*Write $\frac{a}{b}$ instead of $a/b$. The first results in $\frac{a}{b}$, the second results in a much harder to read $a/b$.

*Write $\to$ instead of $->$. The first results in $\to$, the second results in a much harder to read $->$.

*Be consistent in your choice of symbols. You speak of the sets $\mathbb Q$ and $\mathbb R$, but then in the end you speak of the set $Q$. I assume you mean $\mathbb Q$ again, but this is not consistent writing, and will only serve to confuse and anger the reader of your proof.

And finally, and most importantly:
your proof feels like a collection of directionless statements. That is, your proof doesn't really say anything that is particularly untrue, but the problem with the proof is that it doesn't clearly express how the statement you are trying to prove follows from other statements we already know to be true (be they proven statements or axioms).
So, I suggest you rewrite your proof. I would write the proof of like so (with the blanks left for you to fill out, of course)

We wish to prove $A$ is not closed in $\mathbb R$. Remember that if a set is closed, the limit of every convergent sequence contained in the set is also in the set. Therefore, if there exists a convergent sequence in $A$ such that the limit is not in $A$, then $A$ is not closed. We will prove $A$ is not closed by constructing such a sequence.
Let ________ be _________. We therefore define $q_n$ as ___________. We will now prove (1) that $q_n$ is convergent, (2) that $q_n$ is contained in $A$ and (3) that the limit of $q_n$ is not in $A$.
To prove the first point, ___________________
To prove the second point, we note that $q_n\in A$ by definition.
To prove the third point, __________________
Therefore, $q_n$ proves that $A$ is not a closed set.

Note how I was careful to structure the proof in such a way that it is always clear why we are doing something. Why are we constructing $q_n$? Well because we want a sequence with these properties. OK, but why do we need this sequence? Well because such a sequence is not possible if $A$ is closed. Why is this relevant? Well because we want to prove $A$ is not closed, and the existence of the sequence will help with that.
This is very important for the reader. It is why, when writing technical stuff, it's best to follow the approach of:

*

*first, explain what you will do and why you will do the thing.

*then, do the thing.

*then, explain why you did the thing.

A: Constructively:
Let
$$2a_n=10^{-n}\lfloor10^n\sqrt2\rfloor.$$
This sequence is rational and tends to an irrational. (The proof of irrationality of $\sqrt2$ is well kown.)
