This is a homework problem of my real analysis class.
In the previous parts of this problem, I have shown that for a fixed $f \in L^1(\mathbb{R}^n, m^n)$, we have $$\lim_{k \to \infty}k^{n-1}f(kx) = 0$$ for any $x \in \mathbb{R}^n$.
Now I have to prove that for any non-decreasing function $\phi : \mathbb{N} \to [0, \infty)$ with the property $\displaystyle{\lim_{k \to \infty}\phi(k)} = \infty$, the above result fails if $k^{n-1}$ is replaced with $k^{n-1}\phi(k)$.
I currently have no idea at all. Any help would be appreciated! Thank you!