If $a,b\in\mathbb R$ with $a<b$, then there is some rational $r$ with $a<r<b$.
102583
Infinite number of rationals between any two reals.
13550
For an irrational number $\alpha$, prove that the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$
124448
For every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n$ approaches $a$.
47856
$f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
42500
Let $A$ and $B$ be matrices over $\mathbb C$. Then pick out the correct statements.
71012
If $a, b$ are positive integers such that $a|b^2, b^2|a^3,....$ then $a=b$.
27899
Can we prove the existence of $A\cup B$ without the union axiom?
123979
Checking proof for: It is possible to find addition assumptions on the sets $A$ and $B$, so $P(A)\cup P(B) = P(A\cup B)$.
86505
Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$
108305
How can we show that the dyadic rationals are dense in $\mathbb R$?
34154
Prove the statement $\forall x\in\mathbb N (x > 1\to\exists k\in\mathbb N\exists m \in\mathbb N (m \equiv 1 (\text{mod }2) \wedge x = 2^km))$.
151852
How to prove $A \subset B \iff A \cup B = B$
113892
Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
125164
How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
76049
Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$
118239
If $A, B, C$ are sets prove that $(A\setminus B)\setminus C\subseteq A\setminus(B\setminus C)$
50665
If $a,b$ is an $R$-sequence, then $ax-b$ is prime (Eisenbud, Exercise 10.4)