Question

Prove mathematically that if a Turing Machine runs in time O(g(n)), then it runs in time O(h(g(n)) +c),
for any constant c ≥ 0 and any functions g(n) and h(n) where h(n) ≥ n.

Answer 1

f(n) is O(g(n)) then
   there exist a constant c > 0, and a constant n₀ such that for all n ≥ n₀: f(n) ≤ c*g(n)

hence:
   there exist a constant c > 0, and a constant n₀ such that for all n ≥ n₀: g(n) ≥ (1/c)*f(n)
     note that since c > 0, then the constant (1/c) > 0

hence:
   there exist a constant k > 0, namely h = (1/c), and a constant n₀ such that for all n ≥ n₀: g(n) ≥ h*f(n)

which is the definition of g(n) = Ω (f(n)).