Starting from proposition 36 the triangle inequality derive

Starting from proposition 3.6, the triangle inequality, derive the following alternative forms of it: ||upsilon-omega||,||upsilon+omega|| ||upsioln ||omega||, and ||upsilon ||upsilon||-||omega||, for upsilon,omega V, where V is a Euclidean space. (4)

Solution

triangle inequality is

||v|| + ||w|| >= ||v+w||...........................(1)

put -w in the place w in (1), we get

||v|| +||-w|| >= ||v+w||

=>

||v|| +||w|| >= ||v+w||

thus proved

put v+w in the place of v, -w in the place of w in (1), we get

||v+w|| + ||-w|| >= ||v+w-w|| = ||v||

=>
||v+w|| + ||w|| >= ||v||

=>

||v+w|| >= ||v|| -||w||

thus proved

put v-w in the place of v in (1), we get

||v-w|| + ||w|| >= ||v-w+w|| = ||v||

=>

||v-w|| >= ||v||-||w||

thus proved