= Solution
{c}
Take any finite selection of distinct indices $i_1,\ldots,i_r$ and a relation
$$
\alpha_1e^{(i_1)}+\cdots+\alpha_re^{(i_r)}=0.
$$
The coordinate at $i_j$ on the left is exactly $\alpha_j$, while the <zero vector> has coordinate zero. Hence $\alpha_j=0$ for each $j$. Every finite <subset> of $E$ is therefore <linearly independent>, which is the definition of <linear independence> for an infinite <set>. Thus \b[$E$ is <linearly independent>] in the <vector space> of <finitely supported sequences>.
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