Proposition 8

Let $a = (a_i)$ be any point of a product space $X = \prod_{i \in I} X_i$; then the set $D$ of points $x \in X$ such that $\mathrm{pr}_i x = a_i$ except for a finite number of indices $i$ is dense in $X$.

For each $x \in X$ and each elementary set $V = \prod_{i \in I} U_i$ which contains $x$, we have $U_i = X_i$ except for indices $i$ belonging to a finite subset $J$ of $I$; if we take $y_i = x_i$ for $i \in J$ and $y_i = a_i$ for $i \notin J$, it is clear that

hence the result.