Let the sequence of integrable functions $\displaystyle ( f_n )$ be defined on the interval $\displaystyle [0,\infty)$ in which $\displaystyle f_n = \dfrac{1}{n} 1_{[0,n]}$ on $\displaystyle [0,\infty)$.

Prove that there is no integrable function that dominates the sequence.

My work:

Suppose there exist a function$\displaystyle f $that dominates the sequence.

Hence,

$\displaystyle \int_{[0,\infty)} f dm $

$\displaystyle >\int_{[0,n]} f dm$

$\displaystyle >\sum_{k=1}^{n} \int_{[k-1,k)} f dm$

$\displaystyle >\sum_{k=1}^{n} \int_{[k-1,k)} \dfrac{1}{n} dm$

I was stuck here.

Can anyone help me to proceed?

Or if this is not the way to prove,can anyone please show me the correct way?

Thanks.