Prove by cases that $|x + y| \le |x| + |y|$ for all real numbers $x,\,y$.
22237
Proof of triangle inequality
152311
Proving expression$\geq 2$, where variables are $x,y,z$
129584
How prove $E|X|^p<\infty$ and $E|Y|^p<\infty$, if $E|X+Y|^p<\infty$
107774
Prove $\|x\|_1\le \sqrt n \|x\|_2$
13421
Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$
32149
Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
52380
How to prove the function $e^x$ is continous for all real numbers
72313
Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$
27911
How can I prove that $xy\leq x^2+y^2$?
11289
Why $d(x) + d(y) \le n$ when proving Mantel's theorem
39708
Proving $(A\le B)\vee (B\le A)$ for sets $A$ and $B$
36507
If $x,y>0$ and $x^2+y^3\ge x^3+y^4$, prove that $x^3+y^3 \le 2$.
89792
Let $x,y$ be positive real numbers. Then $x<y$ iff $x^n<y^n$ for all $n\in \mathbb{N}$
68456
Prove that there exists irrational numbers $x$ and $y$ such that $x + y$ is rational, without using subtraction
154945
$|px^2+qx+r|\le1$ for all $x$ in $[-1; 1]$, show that $|rx^2+qx+p|\le2$ for all $x$ in $[-1; 1]$
3820
Prove that $||x|-|y||\le |x-y|$