Let $E$ be measurable function, and $\{u_n\}_{n=1}^\infty\cup\{u\}\subset L^1(E)$. Let further $$\int\limits_E |u-u_n|\,dt < \frac{\varepsilon}{2^{n+2}}$$ and $f_n$ be simple functions.
I need to prove that $$\sum\limits_{k=1}^\infty \int\limits_E |u_{k}-u_{k-1}|\,dt < \int\limits_E |u|\,dt + \varepsilon,$$
where $u_0=0$.
I have a problem with solving this problem:
If I represent $|u_k-u_{k-1}|=|u_k-u|+|u-u_{k-1}|$ I won't get the result I need.
If i use the fact that $u=\sum\limits_{k=1}^\infty (u_k-u_{k-1})$, I get the inequality with the opposite sign.
Could you give me some hint?