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?
How do I prove that this integral is positive?
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definite-integrals
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0Isn't that negative? It is decreasing. – 2017-01-25
2 Answers
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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 $\int_0^1f(x+n)-f(n+1)\;dx\geq 0$.
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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 $\int_0^1 f(x+n)\geq\int_0^1f(1+n)$.