From book 'A short course on Banach space Theory by N.L. Carothers'.
My question is How $P_n(z_m) = z_{\mathop{\min \{m, n\}}}$ means that there is a single sequence of scalars $(a_i)$ such that $z_n = \sum_{i=1}^n a_i x_i$ where $a_i$ is independent of $n$, and so $z = \sum_{i=1}^\infty a_i x_i$ and how the uniqueness of the expansion of $z$ with respect to the basis implies $P_n z = z_n$.
If $n\ge m$ then $P_n(z_m)=z_m;\ $ that is $P_n$ projects $z_m$ to itself. Now, the expansion of $z_m$ in the basis $\{x_i\}_{i\in I}$ is of the form $z_m=\sum_{i=1}^\infty a_ix_i$. Applying $P_n:n \geq m$ we see that $a_i=0$ for each $i> m$. Indeed,
$P_n(z_m)=\sum_{i=1}^n a_ix_i=z_m=P_{n+1}(z_m)=\sum_{i=1}^{n+1} a_ix_i\Rightarrow a_{n+1}=0$, and the general result follows by induction.