What's wrong with this inductive proof that mergesort is O(n)?

Question

Comparison based sorting is big omega of nlog(n), so we know that mergesort can't be O(n). Nevertheless, I can't find the problem with the following proof:

Proposition P(n): For a list of length n, mergesort takes O(n) time.

P(0): merge sort on the empty list just returns the empty list.

Strong induction: Assume P(1), ..., P(n-1) and try to prove P(n). We know that at each step in a recursive mergesort, two approximately "half-lists" are mergesorted and then "zipped up". The mergesorting of each half list takes, by induction, O(n/2) time. The zipping up takes O(n) time. So the algorithm has a recurrence relation of M(n) = 2M(n/2) + O(n) which is 2O(n/2) + O(n) which is O(n).

Answer 1

Compare the "proof" that linear search is O(1).

1. Linear search on an empty array is O(1).
2. Linear search on a nonempty array compares the first element (O(1)) and then searches the rest of the array (O(1)). O(1) + O(1) = O(1).

The problem here is that, for the induction to work, there must be one big-O constant that works both for the hypothesis and the conclusion. That's impossible here and impossible for your proof.