Is the following claim true or not if true give a proof if

Question

Is the following claim true or not? If true, give a proof. If not, give a counter-example.

Claim: For all positive, increasing function f, g, and h, if f(n) ∈ O(g(n) + h(n)) then either g(n) ∈ Ω(f(n)) or h(n) ∈ Ω(f(n)) (or both are true). Note that A function f is positive and increasing if f(x) ≥ 0 for all x ≥ 0 and f(x) > f(y) whenever x > y ≥ 0.