Problem
Prove that $\mathbb{F}^\infty$ is infinite dimensional
Solution
Construct the infinite sequence of linearly independent vectors in $\mathbb{F}^\infty$ $(1,0,0,\dots), (0,1,0,\dots), \dots$ where the $j$‘th vector in the sequence has a one in the $j$‘th coordinate and zeros everywhere else.
The existence of such a sequence then implies, by P2A14, that $\mathbb{F}^\infty$ is infinite dimensional.