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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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### 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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### 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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### 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 ...
### 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?