I am currently enrolled in an algorithms course and was learning about upper, lower and tight bounds of functions.
I am confused on how to show that a function $f(n) = O(g(n))$ for some $n > n_0$ and $c$.
The definition for upper bound is: there exists positive constants $c$ and $n_0$ such that $0 \le f(n) \le c \cdot g(n)$ for all $n \ge n_0$.
I am currently stuck on a homework question similar to this:
$$T(n) = 25 n^5 \log(n) + 15n^5 + 8^5$$ Show that $T(n)$ has an upper bound of $O(n^5 \log(n))$
So this is what I have done so far:
$$ 0 \le 25n^5 \log(n) + 15n^5 + 8^5 \le c n^5 \log(n) $$
divide everything by $n^5$
$$ 0 \le 25\log(n) + 15 + 8^5/n^5 \le c\log(n) $$
Now I am stuck, how do I deal with $8^5/n^5$?