$\phi(x) = |x|$ for $x\in [-1,1]$ and $\phi(x+2)=\phi(x)$ for all $x\in \mathbb{R}$. Show $|\phi(t) - \phi(s)|\leq |t-s|$
116056
Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.
47790
If $x \geq 2$ prove that $\sum_{ n \leq x} \frac{1}{\phi(n)} = \mathcal{O}(\log(x))$
121544
Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
62382
I suppose $(a,m)=(a-1,m)=1$, show that $1+a+a^2+\ldots+a^{\phi(m)-1}\equiv0\pmod m$
58663
How to show that if $\phi(n)$ equals to n itself, then n must be 1?
78463
Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$
28822
$|x| - |y| \leq |x-y|?$
28125
Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$
64457
Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$
77026
Proof with inequalities
155232
Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
4512
Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
119118
Prove $n\mid \phi(2^n-1)$
49240
$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert$
78754
Showing that $E[X|X<x]$ is smaller or equal than $E[X]$ for all x
106585
Show $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$.
125582
Let $\phi:G_1\to G_2$ be a group homomorphism. Show $\phi(g^{-1})=(\phi(g))^{-1}$.