Let $f(n)$ and $g(n)$ be positive functions such that $f(n)$ is $O(g(n))$ and $g(n) \ge 1$ for all $n \ge 1$. Using the definition of “big Oh” show that $f(n) + k$ is $O(g(n))$, where $k > 0$ is constant.
Since $f(n)$ is $O(g(n))$ then there are constants $c′ > 0$ and integer n'0 $\ge 1$ such that $f(n) \le c′g(n)$ for all $n \ge $ n′0. Hence, $f(n) + k \leq $ c′ $\ldots$. Also, $g(n) \ge 1$ for all $n \ge 1$.
I'm honestly stuck on this question and would love to receive some help. So far I have this:
$$f(n)+k \le c*g(n)$$
and then $f(n) \le c*g(n)-k$ I'm sure there is a way to get rid of the $k$ somehow, but I can't figure it out and I don't how to show it using the definition of big Oh.
Thanks
Lets take $c = c' + k$ and assume $n \geq n_0$. Then $$f(n) + k \leq c' \cdot g(n) + k \leq c' \cdot g(n) + k \cdot g(n) = (c' + k) \cdot g(n) = c \cdot g(n)$$