How to prove this statement?

Question

Assume you have functions T and S such that T(n) =O(S(n)).

For the following statement, decide whether you think it is True or False.

If you think the answer is True, prove it, otherwise, provide a counter example.

Statement: T(n)+2S(n)=O(S(n))

Answer 1

FALSE

T(n) = n+1

S(n) = n-1

O(n) = n+2

T(n) = n+1 = (n-1)+2 = O(S(n))

T(n) + 2 S(n) = n+1 + 2(n-1) = n+1 + 2n - 2 = 3n - 1

which is NOT O(S(n))