I recently came across a family of infinite graphs (in the context of two-dimensional convexity) that don't have induced 4-paths (paths with 4 vertices).
Note that the complement of a 4-path is again a 4-path.

Clearly, every induced $n+1$-cycle contains an induced $n$-path.

Hence, by the Strong Perfect Graph Theorem of Chudnowski, Robertson, Seymour, and Thomas,
graphs without induced 4-paths are perfect.

Can anyone provide a simple proof of that fact?

Having no induced 4-paths seems like a very strong condition.