How do I prove that this integral is positive?

Question

Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic decreasing positive function. While writing a proof, I came across the following integral: $$\int_0^1f(x+n)-f(n+1)dx$$where $n\in\mathbb{N}$. I know that this integral is positive, but how do I write a rigorous proof of this being positive?

Answer 1

If $x\leq 1$ then $x+n\leq n+1$, hence $f(x+n)\geq f(n+1)$ since $f$ is decreasing. Therefore the integrand is non-negative, so $\int_0^1f(x+n)-f(n+1)\;dx\geq 0$.

Answer 2

For $x\leq1$ then $x+n\leq1+n$ and $f$ is decreasing so $f(x+n)\geq f(1+n)$ or $f(x+n)-f(1+n)\geq 0$ thus $\int_0^1 f(x+n)-f(1+n)\geq 0$ or $\int_0^1 f(x+n)\geq\int_0^1f(1+n)$.