I'm trying to give proofs about this lemma $$\begin{array}{l}{\text { Lemma 5.4. Let }\left\{x_{n}\right\} \text { be a sequence in a Banach space } X .} \\ {\text { (a) } \exists\left\{y_{n}\right\} \subseteq X^{*} \text { biorthogonal to }\left\{x_{n}\right\} \Longleftrightarrow\left\{x_{n}\right\} \text { is minimal. }}\end{array}$$
For $\Leftarrow$ direction, my idea is trying to construct $\{y_n\}$ that is biorthogonal to $\{x_n\}$and my attempt is to use minimal as orthogonality and define $y_n=\frac{x_n}{||x_n||}$ for $n=m$ and $y_n=x_n $ for $n\neq m$. Then $\left\langle x_{m}, y_{n}\right\rangle=\delta_{m n} \text { for all } m, n \in \mathbb{N}$.
I have trouble contructing such $\{y_n\}$ since I can't show $x_m$ is orthogonal to $\overline{\operatorname{span}}\left\{x_{n}\right\}_{n \neq m}$. Is my direction correct? Any help is appreciated.