Path-connectedness is product-closed

Property-theoretic statement
The property of topological spaces of being path-connected is a product-closed property of topological spaces.

Verbal statement
A product (finite or infinite) of path-connected spaces is path-connected in the product topology.

Proof outline
We need to show that any two points in the product can be joined by a path:


 * Start with the two points; write them down as tuples
 * In each coordinate, construct a path from that coordinate for one point, to that coordinate for the other point
 * Take the path, which in a given coordinate, is the path constructed above for that coordinate. The continuity of this follows from the universal property of the product topology