I found a response to a question that confused me. The answer contains the following sequence of steps:
$\max_{y \in X^*} \frac{|y(z)|}{\|y\|_{X^*}} = \max_{y \in X^*} \frac{|y(z)|}{\max_{x \in X, \|x\|_{X}=1} |y(x)|} = \max_{y \in X^*} \max_{x \in X,\|x\|_{X}=1}\frac{|y(z)|}{|y(x)|}$
Why can one just move the max out of the denominator? Consider the following examples where the sups take on different values in $\mathbb{R}$:
$$\sup_{x\in \mathbb{R}} \frac{1}{x}\\ \frac{1}{\sup_{x\in \mathbb{R} }x}$$
(I'm looking at this post)
For $A\subseteq (0,+\infty)$, $\inf\frac{1}{A}=\frac{1}{\sup A}$ (you can find a proof here).
You can show the norm equality much easier like this: $\|i(z)\|=\sup_{y^*\in X^*}\frac{|y^*(z)|}{\|y^*\|}=\|z\|$, where the first equality follows from the definitions of $i$ and the norm in $X^{**}$ and the second one holds from a consequence of the Hahn-Banach theorem which says that the norm of a $z\in X$ is achieved by a functional $y^*\in X^*$ of norm one.