$S_F$ is the space of all real sequences $\mathbf a=(a_n)_{n=1}^{\infty}$ such that each sequence $\mathbf a\in S_F$ is eventually zero.
Show that the following definitions all give norms on $S_F$,
$$\|\mathbf a\|_{\infty}=\max_{n\geq1}\lvert a_n\rvert$$ $$\|\mathbf a\|_1=\sum_{n=1}^{\infty}\lvert a_n\rvert$$ $$\|\mathbf a\|_2=\Bigg(\sum_{n=1}^{\infty}\lvert a_n\rvert^2\Bigg)^{1/2}$$
I'm a bit unsure what I should do here. Do I just prove that all 3 functions are norms based on the properties required for a function to be a norm and the extra information that the sequences are all eventually zero?
PS. Does the fact that the sequences are eventually zero imply that they tend to zero? I would have thought this is the case but I'm not sure if I can assume it.
Start by noting that $S_F$ is a vector space and thus norms can be defined. To show that $||\cdot||\colon S_F\to \mathbb{R}_+$ defines a norm you need to prove that it satisfies the axioms for the norm.
The structure of $S_F$ should make the proofs easier, since all sums are actually finite for a given $x\in S_F$. For example when proving $||\cdot||_2$ is a norm, you can use the fact that for a given sequence $(x_n)\in S_F$ there exists $N\in \mathbb{N}$ s.t. $x_n=0$ for all $n>N$. Thus you can use the properties of the Euclidean norm.
For your PS question: The fact that the sequences are eventually zero means that for all $(x_n)\in S_F$ there exists $N$ s.t. $x_n=0$ for all $n>N$. Thus the convergence to zero is rather trivial.