This surely has been solved in the context of scheduling already! (Shall I ask on some computer SE instead?)

Assume we have a set of closed "intervals" on $\mathbb Z$ ($\mathbb R$ isn't different, I guess) and like to pick the maximum cardinality of nonintersecting intervals from it. I bet the obvious greedy algorithm (first throw out all intervals that completely enclose another, then pick leftmost, repeat) already gives the solution.

A reference would suffice. (If the proof is trivial, please leave it to me :-)