Suppose $U = \{(x_1,x_2,\dots) \in \mathbf{F}^\infty : x_j \ne 0 \text{ for only finitely many }j\}$

1. Show that $U$ is a subspace of $\mathbf{F}^\infty$.
2. Prove that $\mathbf{F}^\infty/U$ is infinite-dimensional.

TODO