Projection maps non-zero element to non-zero element?

Question

The following screenshot is taken from the book 'Topics in Banach Space Theory'.

In the third line of the proof above, I don't understand why $0 \neq y \in Y \Rightarrow S_N(y)\neq0$.

Can't we have the first $N$ entries of the sequence $y$ are zero while the remaining entries can be non-zero. In this case, we have $y \neq 0$ but $S_N(y)=0$ right? Please correct me if I am wrong.

Answer 1

The if not refers to the negation of the statement $\forall\, n\in\mathbb N\;\exists y_n\in Y, \|y_n\|=1 \text{ and } e_k^*(y_n)=0$ for all $k\le n$. Hence there is $N$ such that for all $y=\sum_na_ne_n\in Y$ (with $\|y\|=1$) there is $k\in\lbrace1,\ldots,N\rbrace$ such that $a_k=e_k^*(y)\neq 0$.