How to prove $(\mathbb{R}\backslash \mathbb{Q})\cap (x,y)\neq \emptyset$ for $x,y\in \mathbb{R}$ and $x<y$?
4381
Proof that the set of irrational numbers is dense in reals
65395
Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
92740
prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$
7607
How to prove that $\mathbb{Q}$ is not the intersection of a countable collection of open sets.
108955
Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$
64586
How to prove that $\neg((A\cap B)\cup (\neg A \cap C)) = (A\cap\neg B)\cup (\neg A\cap \neg C)$
61375
Meaning of $\mathbb{R}[x]$
157597
Homeomorphism between $\mathbb{Q}$ and $\mathbb{Q}(>0)$, and $\mathbb{Q}(\ge 0)$
125161
Why is the Cartesian product of $A$ and $\varnothing$ empty?
80892
Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?
17748
Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?
17525
Why the interior of $\mathbb{Q}$ in $\mathbb{R}$ is empty?
27188
Bijection between $\mathbb{R}$ and $\mathbb{R}/\mathbb{Q}$
122944
Prove $|y''(x)| \leq 40$ for all $x \in [1,3]$.
59187
Proof, is $\forall x : (P(x) \lor Q(x)) \Leftrightarrow \forall x : P(x) \lor \forall x : Q(x)$?
65379
Deriving $\neg R$ from $\{R↔(R∨(P∧¬P)), R↔¬P, ¬P→(P↔(Q→Q)), P→Q\}$
54496
Showing that $\mathbb{R}$ and $\mathbb{R}\backslash\mathbb{Q}$ are equinumerous using Cantor-Bernstein