I came across this problem:
Find the mistake of the following inductive proof:
Theorem: all algorithms have the same time complexity.
Proof: (By induction on the number of algorithms.)
The statement is trivial for one algorithm.
Let the statement holds for some K. Consider (K+1) algorithms. Remove any and algorithm, then the remaining algorithms have the same time complexity according to the assumption. Return the algorithm back and remove another one, then the remaining algorithms have the same time complexity again. It means all (K + 1) have the same time complexity.
My attempt: Consider the case of 2 algorithms (K+1) = 2, each with different time complexity. By removing any of them, we have only one algorithm where the statement holds. However, this is not two if each one has different time complexity.
Is this the correct way to solve the problem?