ag.algebraic geometry - Is $Xtimes mathbb{P}^{n}$ connected for a smooth, proper and connected scheme $X$?

This question is taking ridiculously many attempts to get a clean, correct answer.

First: As inkspot points out, a scheme is connected iff it has no idempotent global sections other than zero and one. Thus, whether a scheme is connected is determined purely by its ring of global sections.

Second: As Mattia Talpo and Charles Siegel both observe, $X times_S mathbb{P}^n_S$ is naturally isomorphic to $mathbb{P}^n_X$.

Third: The global section ring of $mathbb{P}^n_X$ is isomorphic to the global section ring of $X$. To see this, first observe that if $f colon mathbb{P}^n_X to X$ is the obvious morphism, then
$$Gamma(mathbb{P}^n_X, mathcal{O}_{mathbb{P}^n_X}) cong Gamma(X, f_{*}mathcal{O}_{mathbb{P}^n_X}).$$
This follows from the definition of $f_{*}$. Thus, it suffices to show that $f_* mathcal{O}_{mathbb{P}^n_X} = mathcal{O}_X$. For this, we can work locally on $X$. If $U subset X$ is an open affine, then our desired statement $f_* mathcal{O}_{mathbb{P}^n_U} = mathcal{O}_U$ is true by standard facts about projective space over an affine scheme.

Conclusion: $X times_S mathbb{P}^n_S$ is connected iff $X$ is. This statement requires no hypotheses on either $X$ or $S$ (not even local Noetherianness).