$\lim_{x \rightarrow \infty} f(x) \not = 0$ $\implies$ $\int_0^{\infty} f(x)dx$ diverges?

Question

What can I use to prove the following:

If $f:[0, \infty[ \rightarrow \mathbb{R}$ integrable in all $[0,c]$, $c>0$ and

if (limit exists and) $$\lim_{x \rightarrow \infty} f(x) \not = 0$$ then $$\int_0^{\infty} f(x)dx$$ diverges.

I understand the idea that the integral must diverge, since the terms of the sequence never go to zero (i.e. the integral or the sum continues accumulating to infinity).

However I'm lost as to what kind of notation to use to display this. I think I don't need Riemann sums in this.

Answer 1

You can follow the following steps:

1. Let $\lim_{x\rightarrow\infty}f(x)=a\neq0$ and then, for a given (small) $\epsilon>0$ there exists $M>0$ such that $f(x)>a-\epsilon$ whenever $x>M$.
2. Separate the integral $\int_{0}^{\infty}f(x) dx = \int_{0}^{M}f(x) dx + \int_{M}^{\infty}f(x) dx$.
3. Use 1. to conclude that the second portion diverges and therefore, the integral diverges.