In a linear vector space that is the Euclidean space $\mathbb{E}_{\infty}$, we have the Cauchy-Schwarz inequality
$$ |\langle x,y \rangle| \leq |x| |y|,$$
where both $x,y \in\mathbb{E}_{\infty}$. Explicitly $x=(\xi_{1},\xi_{2},\cdots)$ and $y=(\eta_{1},\eta_{2},\cdots)$, and we assume here that both are finite in length (i.e. the series for $|x|^{2}$ and $|y|^{2}$ converge).
I am familiar with several proofs of this inequality. However, I am unable to follow the particular approach that Friedman indicates in one of the problems (Problem 1.2, p.6) in his book (also I noticed it's posted online here) where he says that we can prove the inequality by using the result
$$ |\alpha x + \beta y|^{2}=\langle \alpha x + \beta y, \alpha x + \beta y\rangle = \alpha^{2} \langle x,x \rangle +2 \alpha \beta \langle x,y \rangle + \beta^{2} \langle y,y \rangle,$$
which holds for any $\alpha,\beta$ scalars, and by putting
$$ x_{n}=(\xi_{1},\xi_{2},\cdots,\xi_{n},0,0,\cdots), $$ $$ \alpha = |y|^{2}, $$ $$ \beta=\langle x_{n},y\rangle, $$
to prove that $\langle x_{n},y\rangle \leq |x_{n}| |y|$.
How do we proceed using this specific approach (not other approaches) to explicitly reach the inequality?
I expect it's a typo, and $\ \beta\ $ should be $\ -\big\langle x_n, y\big\rangle\ $. You'll then get (with $\ \alpha=|y|^2\ $) \begin{align} \big|\alpha x_n+\beta y|^2&=|y|^4|x_n|^2-2|y|^2\big\langle x_n, y\big\rangle^2+ |y|^2\big\langle x_n, y\big\rangle^2\\ &=|y|^4|x_n|^2-|y|^2\big\langle x_n, y\big\rangle^2\ , \end{align} or \begin{align} \big\langle x_n, y\big\rangle^2&= |y|^2|x_n|^2-\frac{\big|\alpha x_n+\beta y|^2}{|y|^2}\\ &\le |y|^2|x_n|^2\ , \end{align} from which the Cauchy-Schwarz inequality follows immediately.