Example of a function in $L^1[0,1]$

Question

I am trying to find an example of a sequence $f_n$ in $L^1[0,1]$ (Lebesgue integrable functions) such that

1. $ f_n \rightarrow 0 \forall x \in [0,1]$,

2. $\int_0^1 |f_n(x)| dx=1 \forall n \in \mathbb{N}$

3. the sequence $\{ f_n \}$ does not have a limit in $L^1[0,1]$

I have thought of the following example:

$f_n(x)= n$ if $0\leq x<1/n$ and $f_n(x)= 0$ otherwise.

steps 1 and 2 are easy to prove, but I m struggling with step number 3. Can anyone help?

Thanks!

Answer 1

The most direct way to prove that this sequence does not converge is to prove that it is not Cauchy. By a direct computation, you can calculate that, when $n > m$, $\|f_n - f_m\|_{L^1} = 2 - \frac{2m}{n}$. Let $N$ be arbitrary. By setting $m = N$ and $n \gg m$, $\|f_n - f_m\|_{L^1} \approx 2$, so the sequence is not Cauchy.