An inequality

Thursday 18 Nov 2021 in Lemma | 1 min read

Inequalities

Let \(x,y,w,z \in [0,1]\). Then \[ \left| xy-wz \right| \leq \left| x-w \right|+\left| y-z \right| \, . \]

Proof.

\[\begin{eqnarray} \left| xy-wz \right|&=& \left| xy-yw+wy-wz \right|= \left| y(x-w)+w(y-z) \right| \nonumber \\ &\leq& \left| y(x-w)\right| +\left|w(y-z) \right| \nonumber \\ &\leq& \left| x-w\right| +\left|y-z \right| \, . \nonumber \qquad \blacksquare \end{eqnarray}\]