Question first:

Show that if $s_1 < s_2 < \dots$ is an increasing sequence of positive integers and $P(x)$ is a nonzero polynomial then we cannot have $$ P(x) \equiv \prod_{j=1}^\infty (1 - x^{s_j}) $$ as formal series.

The right-hand side really means $\lim_{N \to \infty} \prod_{j=1}^N (1 - x^{s_j})$ where the notion of convergence is as described in this math.SE answer by Bill.

The intuition is that if we take $x \to 1^-$ then the left-hand side tends to zero at a slower rate than the right-hand side, since $P$ has the root $1$ with only finite multiplicity, while every factor on the RHS tends to zero. However, since *formal* and *functional* power series aren't actually the same thing (see link above), I'm not sure how to make this precise, even after formally inverting the $1-x$ factors on the left-hand side (by multiplying both sides by $(1+x+x^2+\dots+)^k$).

Does anyone know the correct incantations?

(For context, this came up in a solution to a recent IMO proposal. Since it was for high school students I swept the convergence issues under the rug, but myself I'd like to know exactly what the right thing to do is.)