Understanding Theorem 12.38 Bruckner's Real Analysis The following is a theorem from Bruckner's Real Analysis:

How the underlined formulas (red and green) hold?

Red: We can write $y_1 = \frac{\alpha}{h_0} \left( \frac{h_0}{\alpha} y + x_0 \right)$ and estimate its norm as follows:
$$||y_1|| = \left| \frac{\alpha}{h_0} \right| \left| \left| \left( \frac{h_0}{\alpha} y + x_0 \right) \right| \right| \geq \left| \frac{\alpha}{h_0} \right|  \inf_y\left| \left| \left( \frac{h_0}{\alpha} y + x_0 \right) \right| \right| = \left| \frac{\alpha}{h_0} \right| h_0 = |\alpha|$$
Green You already have that $||y^*(y_1)|| = |\alpha| \leq ||y_1||$. If you pick $\alpha = ||y_1||$, you know equality holds, and after normalisation you get the desired equality.
