US coins (, , , ) form a "canonical" coin system. Each coin is worth at least twice the next smaller coin (approximately). It means: any amount you could make with multiple smaller coins, you can make with fewer larger coins. Taking the largest coin first never blocks a better solution.
Proof sketch: if you have two dimes instead of one quarter, replacing them with a quarter plus a nickel gives the same value with fewer coins. The greedy choice is always safe.