This question concerns teaching a proof of the theorem that if a polynomial $f \in k[x]$ over an infinite field $k$ is the zero function (i.e. $f(a) = 0$ for all $a \in k$) then it is also the zero polynomial (i.e. the coefficients of $f$ are all zero). (Note that over a finite field, one can have nonzero polynomials that are zero functions.)
Proof. We will use the fact that a nonzero polynomial in $k[x]$ of degree $n$ has at most $n$ distinct roots, that is, a finite number of roots. Since $f$ is the zero function (condition), $f(a) = 0$ for all $a \in k$ (definition). However, since $k$ is infinite (condition), $f$ has infinitely many roots and thus is the zero polynomial. Were any of the polynomial coefficients nonzero, this would lead to $f$ having a finite number of roots. However, this would contradict $f$ having infinitely many roots and hence there cannot be any nonzero coefficients.
What could be done to improve the clarity of the exposition so that the logic of the proof is easier to follow?