Prove that the series $\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$ converges Let $f$ be a non-negative decreasing function on $[1,+\infty)$. Prove that the series 
$$\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$$
converges.

HINT:
Note that
$$f(n+1) \leq \int_n^{n+1} f(x)dx \leq f(n)$$
