I have the following definition of the projection on product topology:
Let $\{X_i\}_{I\in I}$ be a family of topological spaces and for $j\in I$ let $p_j:\prod\limits_{I\in I}X_i\rightarrow X_j$ be the projection onto the $j$th factor i.e. $p_j((x_i)_{I\in I})=x_j$. Then
For any $j\in I$ the function $p_j$ is continuous.
A function $f:Y\rightarrow\prod\limits_{i\in I}X_i$ is continuous if and only if the composition $p_j\circ f:Y\rightarrow X_j$ is continuous $\forall j\in I$.
However, I'm looking for a proof of this. Everything I've looked at gave part 1. as defined in product topology, but I have yet to see a proof of it (I imagine part 2. follows).
If anyone could provide a proof, that would be great.