compare summation and integral

Question

If we know $f(x)$ is monotonic decreasing on the interval $a \leq x < \infty$, could we obtain following relation formally?

\begin{equation} \int_{m}^{n+1} f(x) dx \leq \sum_{j=m}^n f(j) \leq \int_{m-1}^n f(x)dx \end{equation}

Answer 1

Hint: $\int_{j}^{j+1} f(x)dx \leq f(j) \leq \int_{j-1}^{j}f(x)dx$ and let $j$ runs from $m$ to $n$ and add up.