Asymptotic growth rate of $\sum_{n\leq x}{f(n)}$

Question

I would like to prove that if $f$ is a positive monotone decreasing function then there exists $c>0$ such that $$\sum_{n_{0}\leq n\leq x}{f(n)}=\int_{n_{0}}^{x}{f(t)\,dt}+c+O(f(x))$$ I already proved that $$\sum_{n_{0}\leq n\leq x}{f(n)}=\int_{n_{0}}^{x}{f(t)\,dt}+c+o(1)$$ but I don't know how to improve this bound.

Answer 1

Hint: $\sum_{y \leq n \leq x} f(n) = \int_{x}^y f(t) dt + O(|f(x)| +|f(y)|)$