EXERCISES
8.1.1. Let $x, y, z \in \mathbb{R}^n$.
a) If $||x - z|| < 2$ and $||y - z|| < 3$, prove that $||x - y|| < 5$.
b) If $||x|| < 2$, $||y|| < 3$, and $||z|| < 4$, prove that $||x \cdot y - x \cdot z|| < 14$.
c) If $||x - y|| < 2$ and $||z|| < 3$, prove that $||x \cdot (y \times z) - y \cdot (x \times z)|| < 6$.
d) If $||2x - y|| < 2$ and $||y|| < 1$, prove that $|||x \cdot y||^2 - x \cdot x|| < 2$.
e) If $n = 3$, $||x - y|| < 2$, and $||z|| < 3$, prove that $||x \times z - y \times z|| < 6$.
f) If $n = 3$, $||x|| < 1$, $||y|| < 2$, and $||z|| < 3$, prove that $||x \cdot (y \times z)|| < 6$.
