Projection from Product Topology is Continuous/General Result/Proof

Proof
By definition of the product topology on $S$:
 * $\tau$ is the initial topology on $S$ with respect to $\family {\pr_i}_{i \mathop \in I}$

By definition of the Initial Topoplogy:Definition 2:
 * $\tau$ is the coarsest topology on $S$ such that each $\pr_i: S \to S_i$ is a $\struct{\tau, \tau_i}$-continuous.