Modified sign function VC dimenson

If we have $ f:\mathbb{R} \rightarrow \{\pm 1\}$ , and $ \mathcal{F}$ and $ \mathcal{F}’$ , what are the VC dimensions of

$ \mathcal{F} = \{sign(\prod_{i=1}^n (x-\theta_i), \forall a_i \in \mathbb{R} \}$

$ \mathcal{F}’ = \cup_{n=1}^n \mathcal{F}$

I think VC dimesion of $ \mathcal{F}$ is $ n$ and $ \mathcal{F}’$ is infinity

For $ \mathcal{F}$ , we know that it is in 1D, and expanding the polynimals in makes it possible to separate the points with a large polynomial

For $ \mathcal{F}’$ taking the union will add at least one in each increment, and there are infinite functions. Thus infinity.

Do I have the right approach?