So $F$ is an arbitrary field, and $F[x]$ denotes the set of of formal polynomials with coefficients in $F$. And $A=\{f_i \mid i\geq 1\}$.
I need to show two things,
If $A$ is such that $deg (f_i) \neq deg (f_j)$ for $ i\neq j$ then $A$ is linearly independent.
If $A$ also satisfies that $\{deg(f_i) \mid f_i\in A\} = \mathbb{N}$ then $A$ is a basis for $F[x]$.
So for 1, I'm using coordinates in the standard basis for $F[x]$ and showing that those are clearly independent...but appealing to coordinates isn't something I particularly enjoy, especially since in this case it seems pretty circular given that the standard basis for $F[x]$ is such an $A$.
And while 2 makes total sense intuitively, I'm not sure how to show that $A$ spans...at least not with the precise formalism that proving it entails.