Linear Transformations, Rank-Nullity and Quotient Spaces: Examples and Counterexamples

The two displayed clauses define one function $Sx:\mathbb N\to F$ for each $x$: they specify coordinate $0$, and every other natural coordinate is a successor.

For $a,b\in F$ and $x,y\in F^{\mathbb N}$, both $S(ax+by)$ and $aSx+bSy$ have coordinate $0_F$ at $0$. At a successor $\sigma(n)$ they both have value $ax(n)+by(n)$. Hence the functions are equal pointwise, and $S$ is linear.

If $Sx=Sy$, then for every $n\in\mathbb N$ evaluation at $\sigma(n)$ gives $x(n)=y(n)$. Thus $x=y$ pointwise, so $S$ is injective.

Let $e\in F^{\mathbb N}$ have $e(0)=1_F$ and $e(n)=0_F$ for $n\ne0$. Every $Sx$ has value $0_F$ at coordinate $0$, so $Sx\ne e$ by $0_F\ne1_F$. Therefore $e$ has no preimage and $S$ is not surjective.

The preceding three steps show that $S$ is a linear injection which is not a surjection.