Example of a pointwise bounded sequence of measurable functions

Question

I am supposed to find an example of a pointwise bounded sequence of measurable functions $\left \{ f_{n} \right \}$ on $[0, 1]$ such that each $f_{n}(x)$ is a bounded function but $f^{*}(x)=\lim_{n\rightarrow \infty} \sup f_{n}(x)$ is not a bounded function.

Can anyone help?

Answer 1

Hint: Take $f(x):=\frac{1}{x}$ and consider the truncations $f_n=f\chi_{[1/n,1]}$.