Consider the following proposed proof by contradiction that for all positive integers n, 3n > 2.

Proof by Contradiction: Suppose that the statement is false.

Since n is a positive integer, n ≥ 1. That means 3n ≥ 3 > 2. Thus the statement is true. The fact that the statement is true contradicts the assumption that the statement is false. Thus, the assumption that the statement was false must have been false. Thus, the statement is true.

Select all correct responses.

O The proof contains a simple direct proof, wrapped inside the unnecessary logical packaging of a proof by contradiction framework.

O The proof is not technically incorrect, but it is nevertheless an example of extremely bad proof writing.

O This is what the lecture calls a "fake proof by contradiction"

The proof contains a simple direct proof, wrapped inside the unnecessary logical packaging of a proof by contradiction framework.

Step-by-step explanation:

The proof is rigourous and well written, so we discard the second answer.

This is not a fake proof by contradiction: it does not have any logical fallacies (circular arguments) or additional assumptions, like, for example, the "proof" of "All the horses are the same color". It is factually correct, but it can be rewritten as a direct proof.

A meaningful proof by contradiction depends strongly on the assumption that the statement to prove is false. In this argument, we only this assumption once, thus it is innecessary. Other proofs by contradiction, like the proof of "The square root of 2 is irrational" or Euclid's proof of the infinitude of primes, develop a longer argument based on the new assumption, but this proof doesn't.

To rewrite this without the superfluous framework, erase the parts "Suppose that the statement is false" and "The fact that the statement is true contradicts the assumption that the statement is false. Thus, the assumption that the statement was false must have been false. Thus, the statement is true."