1. ## Lebesgue Integral

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.

2. If there existed such a dominating function, you could apply Lebesgue Dominated Convergence theorem, but

$\displaystyle \lim_nf_n=0$ pointwise (even uniformly!)

and

$\displaystyle \int f_n=1$ for each $\displaystyle n$

Hence we can't interchange limit and integral, a contradiction with having the condictions to apply Lebesgue theorem.