= Solution
Use the <Euclidean inner product> on $\mathbb R^n$, with $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$. If $b=0$, both sides of the desired inequality are zero. Otherwise take the single <unit vector> $v=b/\|b\|$ in part (a). Then
$$
\|a\|^2\geq\frac{\langle a,b\rangle^2}{\|b\|^2},
$$
so
$$
\boxed{\left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)\geq\left(\sum_{i=1}^n a_ib_i\right)^2.}
$$
This is the real <Cauchy-Schwarz inequality>. Equality holds when the two <vectors> are <linearly dependent>.
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