Let $\mathfrak{p}_1, \dotsc, \mathfrak{p}_k$ be relevant homogeneous primes ideals in the graded ring $R := \Bbbk[x_0, \dotsc, x_n]$, where $\Bbbk$ is a field. Prime avoidance (in Eisenbud's terminology) tells us that there exists a nonconstant homogenous polynomial $f \not\in \cup_i \mathfrak{p}_i$. Is there an easy way to see that for some $d \geq 1$, there exist homogeneous $f \in R_d$, $g \in R_{d+1}$, neither of which is contained in $\cup_i \mathfrak{p}_i$?

Note: If $\Bbbk$ is infinite, this is easy: we can use the fact that no vector space is a finite union of proper subspaces to find $\alpha \in R_1 \smallsetminus \cup_i \mathfrak{p}_i$, and then take $f=\alpha$, $g = \alpha^2$.

Motivation: If the $\mathfrak{p}_i$ are the associated primes of a closed subscheme $X$ of $\mathbb{P}^n_{\Bbbk}$ then we can pull back $g/f$ to a nice rational section of $\mathcal{O}_X(1)$. Consequently, we know that $\mathcal{O}_X(1)$ comes from a Cartier divisor. Using the fact that any line bundle can be twisted to a very ample line bundle, this shows that the Cartier divisor group surjects onto $Pic(X)$.