real analysis - Show that the following definitions all give norms on $S_F$

$S_F$ 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.