Let $F_n(t)=\frac{1}{n} \sum_{k=1}^n 1_{X_k \leq t}$ be the emprical distribution function of i.i.d. random variables $X_k\sim U[0,1]$. Define for $t\in [0,1)$ $$M_n(t)=\frac{F_n(t)-t}{1-t}.$$ I have shown that this is a martingale. If anybody is interested, I'm posting it. Now I want to show that if we put $M_n(t)=1$ for $t\geq 1$, then $M_n$ is not UI, but bounded in $L^1$. But then, I am not sure what $\lim_{t\rightarrow 1} M_n(t)$ is, can you help me with that? I guess, I need that to show that it's not UI, right? Furthermore, I'm really stuck with the boundedness, none of the inequalities I found, lead anywhere. Any ideas?
Thank you very much!