1. The following Big-Oh proofs are incorrect and do not sufficiently prove that the given f(n) is in the specified (g(n)). For each proof, state how it is incorrect in one short sentence. You should not suggest a correction.

Note that the claims these proofs are trying to prove are true. Only the proofs themselves are incorrect.

a. Let f(n)=n3+n. We want to prove that f(n)(n3). By the definition of Big-Oh, f(n)(n3) iff f(n)cn3 for some c>0 and all nn0. Using the definition of f(n), n3+ncn3. Let c=2 and n0=1. By substituting in, we have n3+n2n3. So, subtracting an n3 from both sides, we get nn3. Dividing by n for both sides, we get 1n2, which is true for all nn0. So, f(n)(n3).

b.We want to prove that 4n3 is in (n3). Let c4 and n0=1. Then we know 4n34n3, so, 4n3cn3 for any value of nn0. By definition of (n3), 4n3(n3).

c. We want to disprove that n4 is in (n3). For this to be true, it must be the case that n4cn3 for some c>0. But n4n3 can never be true because the 4 exponent causes n4 to always be larger than n3, thus n4(n3).

Please show that why these proofs are INCORRECT.