I'm reading the paper Hypergraphs, Entropy, and Inequalities by Ehud Friedgut. In it, he proves the Cauchy-Schwarz inequality
$$\left(\sum_k a_k b_k\right)^2 \leq \left(\sum_k a_k^2\right)\left(\sum_k b_k^2\right)$$
but the proof makes use of the fact that the vectors $\vec{a}, \vec{b}$ are (positive) integer-valued. He then writes that
The case of real values can be deduced by approximating reals with rationals and deriving the rational case from the integral one.
I'm hoping for some help understanding how the rational case can be derived from the integral one. My intuition is that for finite-dimensional vectors in $\mathbb{Q}^n$, if you look at where the elements lie on the number line, there's always a way to "zoom out" on the number line and find integers with the same spacing as your rational elements, and if the relative distances are the same, then maybe there's some argument that using the inequality on these integers is the same as using it on the original rational vectors. However I'm not sure how this can be formalized.
$\newcommand{\bb}[1]{\left( #1 \right)}$ Suppose $a_k = \frac{u_k}{v_k}$ and $b_k = \frac{x_k}{y_k}$. Let $s = v_1v_2\cdots v_n$, $t = y_1y_2\cdots y_n$. Then: \begin{align*} &\bb{\sum_{k=1}^n a_kb_k}^2 \leq \bb{\sum_{k=1}^n a_k^2}\bb{\sum_{k=1}^n b_k^2} \\ &\iff \bb{\sum_{k=1}^n \frac{u_kx_k}{v_ky_k}}^2 \leq \bb{\sum_{k=1}^n \bb{\frac{u_k}{v_k}}^2}\bb{\sum_{k=1}^n \bb{\frac{x_k}{y_k}}^2} \\ &\iff s^2t^2\bb{\sum_{k=1}^n \frac{u_kx_k}{v_ky_k}}^2 \leq s^2t^2\bb{\sum_{k=1}^n \bb{\frac{u_k}{v_k}}^2}\bb{\sum_{k=1}^n \bb{\frac{x_k}{y_k}}^2} \\ &\iff \bb{\sum_{k=1}^n st\frac{u_kx_k}{v_ky_k}}^2 \leq \bb{\sum_{k=1}^n \bb{s\frac{u_k}{v_k}}^2}\bb{\sum_{k=1}^n \bb{t\frac{x_k}{y_k}}^2} \end{align*} All the summands involved are now integers, thus we have reduced the rational case to the integer case.