I would like to ask about marked place in Proposition 9 from N.H. Duu, On The Existence of Bounded Solutions for Lotka- Volterra Equations, Acta Mathematica Vietnamica 25(2) (2000), 145-159.
Why limit of positive function which has convergent improper integral and its bounded with derivative equal $0$? I found some similar topic here. What's the point of bounded derivative according to proposition from article?
There are well-known counterexamples where $f$ is positive with a convergent improper integral over, say, $[0,\infty)$ and an unbounded derivative such that $f(x) \not\to 0$ as $x \to \infty$.
Suppose though that the derivative is bounded and there exists $B$ such that $|f'(x)| < B$ for all $x$. Then we must have $f(x) \to 0$ as $x \to \infty$.
Otherwise, we could find $\epsilon > 0$ and a sequence $x_n \to \infty$ such that $f(x_n) \geqslant \epsilon$ and $x_{n+1} - x_n > \epsilon/(2B)$. By the MVT, we have $f(x) = f(x_n) + f'(c_n)(x - x_n)$ with $x_n < c_n < x < x_n + \epsilon$ and
$$f(x) > \epsilon - B\epsilon / (2B)= \epsilon/2$$
Hence,
$$\int_{x_1}^\infty f(x) \, dx > \sum_{n=1}^\infty\int_{x_n}^{x_n+\epsilon/(4B)}f(x) \, dx > \sum_{n=1}^\infty\frac{\epsilon^2}{8B} = +\infty,$$
which contradicts the assumption that the improper integral is convergent.