To show that a recursively defined sequence of functions converges pointwise to $0$.

Question

I'm having some difficulty with the following problem:

Let $f_1:\mathbb{R}\to [0,\infty)$ is a measurable function. For $n\in\mathbb{N}$, define $E_n=\{x\in\mathbb{R}:f_n(x)\geq1/n\}$ and $f_{n+1}=f_n-\cfrac{1}{n}\chi_{E_n}$. Show that $f_n\to0$ pointwise on $\mathbb{R}$.

I can see that for a fixed $x\in\mathbb{R}$, the sequence $f_n(x)$ is a decreasing sequence bounded below by $0$, so I know it is a convergent sequence. However, I'm unable to see why the limit is $0$. Any help with this is appreciated. Thanks for your time.

Answer 1

Let $n$ be the first natural number such that $f_n(x)\geqslant \frac{1}{n}$. Then we also have that $f_n(x)\leqslant \frac{1}{n-1}$, by our choice of $n$. Finally, $f_{n+1}(x)=f_n(x)-\frac{1}{n}\leqslant \frac{1}{n-1}-\frac{1}{n}=\frac{1}{n(n-1)}$.