Question

1. Prove that n + log n = O(n) by showing that there exists a constant c > 0 such that n + log n ≤ cn.

Answer 1

$$n+\log ⁡n\le n+\log⁡(1+n+n^2/2!+n^3/3!)\le$$ $$n+\log( e^n) =n+n=2n,\quad n\in \mathbb{N}$$