Let $f$ be a continuous function on $[0, \infty)$ such that $0\leq f \leq Cx^{-1-\rho}$, where $C$ and $\rho$ are positive constants. Let $f_k(x)=kf(kx)$.
$\textbf{Question}$: Show that $f_k$ does not converge to $0$ uniformly on $[0, \infty)$ unless it is identically 0
My findings so far: $f_k(x)$ converges point-wise to $0$, and in fact by cauchy criterion for uniform convergence the convergence is uniform on $[r, \infty)$ for any $r > 0$. But I am quite clueless after this. In particular I am having trouble handling the inequality when trying to show the convergence is not uniform.
This is not a homework problem, any help/ hint is welcome.