Let $f$ be a real decreasing function (resp. $g$ a real increasing function), and $(f_n)$ be a sequence of real decreasing functions (resp. $(g_n)$ be a sequence of real increasing functions) such that $$ \vert f(x)-f_n(x) \vert \leq \frac{C(x)}{\sqrt{n}}, \quad \vert g(x)-g_n(x) \vert \leq \frac{C'(x)}{\sqrt{n}}$$ where $C$ and $C'$ are two (unspecified) functions of $x$. let $k$ be some positive integer. Define $\tau=\min\left\{x \in \left\{1, \dots, k \right\}, f(x) \leq g(x)\right\}$ and $\tau_n=\min\left\{x \in \left\{1, \dots, k \right\}, f_n(x) \leq g_n(x)\right\}$, where the minima are assumed to exist ($\tau$ is the first integer for which f becomes smaller than g, similarly for $\tau_n$). I have two questions:
- Is it possible to leverage the upper bounds on $|f(x)-f_n(x)|$ and $|g(x)-g_n(x)|$ to upper bound $|\tau-\tau_n|$ by a (non-trivial) function of $n$ ?
- What is the smallest $n$ such that $\tau_n=\tau$ ?
Answering the first question should be enough to answer the second, because $|\tau-\tau_n| <1$ implies $\tau=\tau_n$ and we would just need to solve a straightforward equation for $n$.
This question Rate of convergence of roots appears closely related to my problem, but no complete answer was given. My problem is essentially a discretized version of the problem of identifying the rate of convergence of the root of $f_n-g_n$ towards the root of $f-g$.