[tex]a) {x}^{2} + {y}^{2} \geqslant 2xy[/tex]
[tex] {x}^{2} + {y}^{2} - 2xy \geqslant 0[/tex]
[tex] {x}^{2} - 2xy + {y}^{2} \geqslant 0[/tex]
[tex] {(x - y)}^{2} \geqslant 0 \: \: \: \forall \: x,y\:\in\:\mathbb{R}[/tex]
[tex]b) {x}^{2} + {y}^{2} + {z}^{2} \geqslant xy + xz + yz [/tex]
[tex] {x}^{2} + {y}^{2} + {z}^{2} - xy - xz - yz \geqslant 0 \: | \times 2[/tex]
[tex]2 {x}^{2} + 2 {y}^{2} + 2 {z}^{2} - 2xy - 2xz - 2yz \geqslant 0[/tex]
[tex] {x}^{2} + {x}^{2} + {y}^{2} + {y}^{2} + {z}^{2} + {z}^{2} - 2xy - 2xz - 2yz \geqslant 0[/tex]
[tex] {x}^{2} - 2xy + {y}^{2} + {x}^{2} - 2xz + {z}^{2} + {y}^{2} - 2yz + {z}^{2} \geqslant 0[/tex]
[tex] {(x -y )}^{2} + {(x - z)}^{2} + {(y - z)}^{2} \geqslant 0 \: \: \:\forall \: x,y,z\:\in\:\mathbb{R}[/tex]
