Prove Asymptotic Notations of Various kinds

Question

I have a few exercise problems for my Algorithms Home-work and I can't seem to figure out on how to proceed with the proofs of the following relations: (Note that some of them are not necessarily true in which case we need to disprove them using a counter example)

1) Prove that O(O(f(n))) = O(f(n)) (This one seems intuitively correct, but I don't have the mathematical sophistication to prove it

2) Prove that O(Theta(f(n))) = O(f(n))

And I have similar relations like so, that I have to prove or disprove. If anyone can help me in approaching at least one proof among the above, I think I can use the same approach for the rest of them.

Any help will be appreciated!

Answer 1

Your first question is a duplicate of:

What does O(O(f(n))) mean?

The second question should become straightforward once you understand the answer for the first question.