Find a sequence in $L^1$ that satisfies the given properties

Question

Does there exist a sequence of functions $\{f_n\}\in L^1([0,1])$ satisfying the following three properties:

(i) $f_n(x)\to 0$ for any $x\in[0,1]$;

(ii) $\int_{[0,1]}f_nd\mu\le 2013$ for any $n\in\mathbb{N}$;

(ii) $\{f_n\}$ does not converge in $L^1([0,1])$?

Justify your answer.

I am stuck to create an example that satisfies all the properties. For example $f_n(x)=\frac{x}{n}$ does not satisfy all the properties.

Any help would be appreciated. Thanks in advance.

Answer 1

Hint: tall, thin rectangles with constant area.