$\lim\limits_{u\rightarrow \infty}|F_1(u,x)-F_2(u,x)|=0,\forall x\geq 0 \ \implies \ \lim\limits_{u\rightarrow \infty}\sup\limits_{x\geq 0}|F_1(u,x)-F_2(u,x)|=0$ ?
I get the question when I read the proof of Theorem 7 from a paper titled Statistical Inference Using Extreme Order Statistics （link: https://projecteuclid.org/download/pdf_1/euclid.aos/1176343003 ）
Is the statement above always true? If it is not always true, why it is true in the paper's case?
No, take \begin{align*} F_1(u,x) = \frac{(x+1)^2}{u}, \quad F_2(u,x) = \frac{x^2}{u} \end{align*} Clearly, \begin{align*} \lim_{u\rightarrow \infty}|F_1(u,x) - F_2(u,x)| = \lim_{u\rightarrow \infty}\frac{2x + 1}{u} = 0 \end{align*} for all $x \ge 0$. But \begin{align*} \lim_{u\rightarrow \infty}\sup_{x\ge 0}\frac{2x + 1}{u} = \infty \end{align*} You would need $F_1, F_2$ to both be uniformly continuous in $x$.