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### Proof of the Density of Irrationals [duplicate]

So I attempted to prove that between every rational numbers is an irrational as an exercise, and wanted to see if there are problems in my solution. Proof: Suppose $n \in \mathbb{N}$ and $x$ is a ...
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### How to prove $(\mathbb{R}\backslash \mathbb{Q})\cap (x,y)\neq \emptyset$ for $x,y\in \mathbb{R}$ and $x<y$? [duplicate]

How to prove $(\mathbb{R}\backslash \mathbb{Q})\cap (x,y)\neq \emptyset$ for $x,y\in \mathbb{R}$ and $x<y$? Sorry, but I don't even know how to start. Any ideas and impulses?
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### Is complement of a dense set in $\mathbb{R}$ dense in $\mathbb{R}$?

$\mathbb{Q}$ is dense in $\mathbb{R}$. Also, its complement, $\mathbb{R-Q}$, is dense in $\mathbb{R}$. I know that we can proof denseness of $\mathbb{Q}$ and $\mathbb{R-Q}$ separately for each of them....
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### Show that f is discontinuous at every rational and continuous at every irrational. [duplicate]

Let $f(x)=0$ if x is irrational and $f(\frac{p}{q})=\frac{1}{q}$ if $p$ and $q$ are positive integers with no common factors. Show that f is discontinuous at every rational and continuous at every ...
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### What number follows immediately after a rational number?

I recently came across a confusing question on limits and was having trouble solving it. f(x) = \begin{cases} x^2 & \text{if $x$ is rational} \\[1ex] 0 & \text{if $x$ is irrational} \end{...
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### What's between an irrational and rational number? [closed]

There is a rational number between two irrational numbers, and an irrational number between two rational numbers. So what's between an irrational and rational number? I know about rational numbers ...
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### Understanding density of irrational numbers and Archemedian property

From Density of irrationals I know this much of the proof of the density of irrational numbers "We know that $y-x>0$. By the Archimedean property, there exists a positive integer $n$ such ...
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### Dense and nowhere dense

Let $X$ be a topological space and $A$ be a non-empty subset of $X$. Then one can conclude that if $X\setminus A$ is nowhere dense in $X$, $A$ is dense in $X$, Is the above statement true in general?...
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### Question about proof that irrationals are dense in $\mathbb{R}$

I'm looking at Bungo's answer in: Proof that the set of irrational numbers is dense in reals and in the last step it says: "Since $\mathbb{Q}+\sqrt{2}$ is a subset of the irrationals, we conclude ...
### Proof there is an irrational number $r$ in every intervall $a < r < b$ [duplicate]
Proof that for $a,b \in \mathbb{R}$ there is an irrational number $r$ so that $a < r < b$. Basically, proof, that between any two irrationals, there is another irrational r. I'm sure there ...
### Prove: $Cl_{\mathbb{R}}(\mathbb{IQ})=\mathbb{R}$
How to prove: $Cl_{\mathbb{R}}(\mathbb{IQ})=\mathbb{R}$ ($\mathbb{IQ}$-set of irrational numbers)? I know how to prove: $Cl_{\mathbb{R}}(\mathbb{Q})=\mathbb{R}$ ($\ast$) Proof of ($\ast$): Let: \$r \...