Many of the properties of O( ) also hold for the other four asymptotic notions. Prove the following transitivity properties for arbitrary functions f , g, and h: 1. If f(n) = Ω(g(n)) and g(n) =...

Many of the properties of O( ) also hold for the other four asymptotic notions. Prove the following transitivity properties for arbitrary functions f , g, and h:

1. If f(n) = Ω(g(n)) and g(n) = Ω(h(n)), then f(n) = Ω(h(n)).

2.If f(n) = Θ(g(n)) and g(n) = Θ(h(n)), then f(n) = Θ(h(n)).

3.If f(n) = o(g(n)) and g(n) = o(h(n)), then f(n) = o(h(n)).