The following is a problem from Extreme Values, Regular Variation and Point Processes by Resnick.
We say $U \colon (0,\infty) \to (0,\infty)$ is regularly-varying if for all $x>0$, $\lim_{t \to \infty} \frac{U(tx)}{U(t)} = x^{\rho}$ for some $\rho \in \mathbb{R}$.
One of the proposition was to prove that $U$ is regularly varying if there exists $\lambda_n, a_n > 0$ with $\frac{\lambda_{n}}{\lambda_{n+1}} \to 1$ and $a_n \to \infty$ and for all $x>0$, $\lim_{n \to \infty} a_n U(a_nx) = \chi(x)$ positive and finite. Furthermore, in this case $\frac{\chi(x)}{\chi(1)} = x^\rho$.
The exercise is to prove the proposition when the condition "for all $x>0$, $\lim_{n \to \infty} a_n U(a_nx) = \chi(x)$" is relaxed to for all $x$ in some dense set.
We know that $\chi$ on this dense set is monotone, so I believe we can get $\frac{\chi(z_n)}{\chi(1)} \leq \frac{U(ty)}{U(t)} \leq \frac{\chi(x_n)}{\chi(1)}$ for $y$ not in the dense set $z_n < y$ and $z_n$ increasing to $y$, and $x_n > y$ and $x_n$ deceasing to $y$ with $z_n$ and $x_n$ in the set.
But I don't know how to proceed further since I don't know anything about $\chi$.
Thank you.
