How do you prove that 2n is greater than n by mathematical induction?

To prove that (2n > n) for all (n \geq 1) using mathematical induction, we start with the base case: for (n = 1), (2(1) = 2) and (1 < 2), so the statement holds. Next, assume the statement is true for some integer (k), meaning (2k > k). For the inductive step, we need to show that (2(k+1) > k+1). This simplifies to (2k + 2 > k + 1), which can be rearranged to (2k > k - 1). Since the induction hypothesis states (2k > k), we have (2k > k - 1) for (k \geq 1). Therefore, by induction, (2n > n) holds for all (n \geq 1).