These are Big-Oh proofs and use just a short sentence to describe what's wrong with it.

1.1. = = . Let f(n) = n + n. We want to prove that f(n) 0(n). By the definition of Big-Oh, f(n) 6(n) iff f(n) sc.n for some c>0 and all nz no: Using the definition of f(n), n +nscon. Let c = 2 and no = 1. By substituting in, we have n +ns 2n. So, subtracting an n from both sides, we get ns n'. Dividing by n for both sides, we get 1 s n?, which is true for all n2 no: So, f(n) E O(n?). We want to prove that 4n3 is in 0(n). Let c 2 4 and no 1. Then we know_4n s 4n, so, 4.n scnfor any value of n 2 no. By definition of 0(n), 4n3 0(n). n 1.2. . 1.3. C. We want to disprove that n4 is in 0(n?). For this to be true, it must be the case that nt sc.n3 for some c>0. But n s n? can never be true because the 4 exponent causes n4 to always be larger than n, thus n4 & 0(n)