Given numbers $$$1$$$ to $$$n$$$, we have to make all those number equal to zero by choosing some subset and subtracting same value from all of them, we have to use minimum number of operations.

My idea was to always split it into two parts and get two {$$$1,2..n/2-1,n/2$$$} sets from {$$$1,2,..n$$$} set and continue like this recursively. This approach is correct and leads to $$$ceil(log_2(n))$$$ solution. What is the proof this being optimal?