(i) State and prove the Cauchy-Schwarz inequality for vectors in $\mathbb{R}^{n}$. Deduce the inequalities

$$|\mathbf{a}+\mathbf{b}| \leqslant|\mathbf{a}|+|\mathbf{b}| \text { and }|\mathbf{a}+\mathbf{b}+\mathbf{c}| \leqslant|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|$$

for $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^{n}$.

(ii) Show that every point on the intersection of the planes

$$\mathbf{x} \cdot \mathbf{a}=A, \quad \mathbf{x} \cdot \mathbf{b}=B$$

where $\mathbf{a} \neq \mathbf{b}$, satisfies

$$|\mathbf{x}|^{2} \geqslant \frac{(A-B)^{2}}{|\mathbf{a}-\mathbf{b}|^{2}}$$

What happens if $\mathbf{a}=\mathbf{b} ?$

(iii) Using your results from part (i), or otherwise, show that for any $\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{y}_{1}, \mathbf{y}_{2} \in \mathbb{R}^{n}$,

$$\left|\mathbf{x}_{1}-\mathbf{y}_{1}\right|-\left|\mathbf{x}_{1}-\mathbf{y}_{2}\right| \leqslant\left|\mathbf{x}_{2}-\mathbf{y}_{1}\right|+\left|\mathbf{x}_{2}-\mathbf{y}_{2}\right|$$