A locally best choice needs a global proof
A greedy algorithm commits to a choice without revisiting it. Optimal substructure alone is insufficient; you also need an argument that some optimal solution contains the greedy choice.
For maximum-cardinality nonoverlapping interval scheduling, choose the interval finishing earliest. Given an optimal schedule whose first interval finishes later, replace that first interval with the earliest-finishing one. The replacement cannot remove room for subsequent intervals, so an equally good optimum begins with the greedy choice. Repeating the argument solves the remaining suffix.
Contrast coin denominations [1,3,4] and amount six. Taking the largest coin first yields 4+1+1, while 3+3 uses fewer coins. The locally largest denomination has no general optimality guarantee.
Exercise
Write the interval exchange argument in your own words, explicitly stating that the objective is number of intervals, not total weight. Create a weighted example where choosing earliest finish loses to a more valuable overlapping interval.
Check: test a candidate greedy rule against small exhaustive solutions before relying on intuition. A passing test set supports implementation confidence but does not replace the proof for all inputs.