Why Weighted Needs DP

Activities: [($1$, $3$, $2$), ($2$, $5$, $4$), ($4$, $6$, $4$)]. Greedy by end time: (1,3) then (4,6). Total: $2$ + $4$ = $6$. Alternative: just (2,5). Total: $4$. Worse. Greedy wins again. Let me find a real counterexample..

Activities: [($0$, $4$, $3$), ($3$, $5$, $5$), ($4$, $7$, $2$)]. Greedy: (0,4) then (4,7). Total: $5$. Alternative: (3,5) alone. Total: $5$. Tie. The point: with weights, the greedy choice property can fail. Use DP for weighted cases.