Show that for any reals $a; b; x; y \ge 0, (ax + by)^2 \le (a^2 + b^2)(x^2 + y^2)$. (Hint: Use Cauchy-Schwarz inequality in $\mathbb{R}^{2}$.)
132098
Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
64955
Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
7775
Show that $|x-y|\leq |x|+|y|$.
119668
How prove this inequality: $\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|$?
29679
Prove: $|x-y|\leq |x|+|y|$
86209
Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$
118613
Proving an inequality: $|1-e^{i\theta}|\le|\theta|$
35001
Prove inequality using Schwarz's Lemma
81742
Integral inequality (Cauchy-Schwarz)
2504
Inequality $\left|\,x_1\,\right|+\left|\,x_2\,\right|+\cdots+\left|\,x_p\,\right|\leq\sqrt{p}\sqrt{x^2_1+x^2_2+\cdots+x^2_p}$
112676
Inequality $(a+b+c)(ab+bc+ca)(a^3+b^3+c^3)\le (a^2+b^2+c^2)^3$
155232
Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
104498
Show that $x^3y^3(x^3+y^3) \leq 2$
89170
How prove this inequality $\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}\ge x^2+y^2+z^2$
95030
Proving the Cauchy-Schwarz inequality by induction
149889
Prove that $|\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$
12781
Prove the following inequality $\sqrt[n]{x_1x_2...x_n}\ge (n-1)$