Show that such $x$ and $y$ exist
70463
Let $p$ be prime and $a$ be a natural not divisible by $p$. Then, $\exists$ integers $x,y$ such that $ax\equiv y \pmod{p}$ with $0<|x|,|y|<\sqrt{p}$
125209
Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
158882
Prove the existence of some weird family of sets
103451
Show that $Y$ is locally compact.
1027
Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$
21891
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
148585
Show that, for any $\epsilon>0$, there exist two rationals such that $q < x < q'$ and $|q-q'|<\epsilon$
90970
How to prove the line exist?
143072
Proving $|\sin x - \sin y| < |x - y|$
20467
Show that $(y_n)\in l^q$
152495
Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent
124234
Prove that if $X$ and it's closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected.
45481
Prove or disprove validity: $(\forall x \exists y (P(x) \supset Q(y))) \supset(\exists y \forall x (P (x) \supset Q(y)))$
143735
Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$
13656
Prove existence of a real root.
79281
Let $f : X \to Y$ be a function and $E \subseteq X$ and $F \subseteq X$. Show that in general
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.