To show that a recursively defined sequence of functions converges pointwise to $0$.
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, $f_{n+1}(x)=f_n(x)-\frac{1}{n}\leqslant \frac{1}{n-1}-\frac{1}{n}=\frac{1}{n(n-1)}$.