I have to prove the following problem:
Let ($\Omega,F,\mu)$ be a measure space and $f$ a non-negative measureable function $f:\Omega\rightarrow \mathbb{R},$ and $f_n:\Omega\rightarrow \mathbb{R}$ with $x\mapsto \frac{1}{2^n}\max\{ \lfloor 2^nf(x)\rfloor,n\}$.
Show $f_n\leq f_{n+1} \quad \forall n\in \mathbb{N}$
But there is a problem: What if $f≡0$? Then for $n=2$ we get $f(x)=\frac{2}{4}=\frac{4}{8}>\frac{3}{8}=f(x)$ for $n=3$. Doesn't that contradict the statement I have to show?
Also I have to show $\lim f_n= f$ pointwise. But I stopped because of the first problem. Thanks in advance.