Find mistake in the proof by induction of the statement: All people speak the same language. Base case: for n = 1 the statement holds since there is only one person in the group. Induction step: Assume that the statement is true for some natural k. That is, any k people speak the same language. We need to prove the statement for k + 1. Consider a set containing k+ 1 people. Remove any person, say, X from this set. By assumption, the remaining k people speak the same language. Next we return X to the set, and remove another one, say Y. Using induction assumption again, we can show that X speaks the same language as the remaining k - 1 people. We can now return Y to the set, and claim that all of k+1 people speak the same language, as needed. The proof is complete. O The base case is proven incorrectly. O The induction assumption is stated incorrectly. O The proof of induction step does not work for k=2.