Show that

$$\sum_{i=1}^{n} a_{i} b_{i} \leqslant\left(\sum_{i=1}^{n} a_{i}^{2}\right)^{1 / 2}\left(\sum_{i=1}^{n} b_{i}^{2}\right)^{1 / 2}$$

for any real numbers $a_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{n}$. Using this inequality, show that if $\mathbf{a}$ and $\mathbf{b}$ are vectors of unit length in $\mathbb{R}^{n}$ then $|\mathbf{a} \cdot \mathbf{b}| \leqslant 1$.