I want to prove a convergence result as simple as possible. Using a straight forward approach I can prove the result, but I'm 100% sure that there must be a much simpler (and shorter) argument using e.g. functional analysis.
The situation is the following: for $T>0$ we have a series of monotone increasing step functions $(X_N)_N$ with $X_N:[0,T]\to[0,1]$, $X_N(0)=0$ and $X_N(T)=1$. Furthermore, the last jump of $X_N$ occurs before time $T$ (think of $X_N$ as piecewise constant deterministic function with $N$ jumps).
We also have a monotone increasing continuous function $X:[0,T]\to [0,1]$ also satisfying $X(0)=0,X(T)=1$. We know for all $0\le t<T$, that $$\lim_{N\to\infty}\sup_{0\le s\le t}|X_N(s)-X(s)|=0.$$ What we want to prove is that also $$\lim_{N\to\infty}\sup_{0\le s\le T}|X_N(s)-X(s)|=0.$$
Does anybody know some functional analysis theorem stating such results or any literature I can check?