Show convergence of improper integral with nearest integer function

Question

Suppose $f$ is decreasing and $\lim_{x\rightarrow\infty}f(x)=0$.

Then why $$\int_0^\infty(-1)^{[x]}f(x)dx$$ converges? ($[x]$ is the nearest integer function). Any hint?

Answer 1

(More than a) Hint:

$$\int_0^\infty (-1)^{[x]}f(x)\,dx = \int_0^{1/2} f(x)\,dx + \sum_{k=1}^\infty (-1)^k \int_{k-1/2}^{k+1/2} f(x)\,dx.$$

Answer 2

Hint. First of all it is easy to show that $$ F(n)=\sum_{k=1}^n\int_{k-1/2}^{k+1/2} (-1)^{[x]}f(x)\,dx, $$ converges as according to the alternating series test.