Sequence of function in $L^p$

Question

I used following state when I solved some problems.

But I can't prove that state. How can I prove that?

Let $1<p<\infty,$ $f_k\in L^p$ for $k\ge 1$ such that

1. There is an $a$ with $\|f_k\|_p \leq a$ for $k=1,2,...$

2. $\lim f_k(x)=:f(x)$ exists a.$\,$e.

Then $f\in L^p$ (i.$\,$e. $\|f\|_p < \infty$).

Answer 1

Hint

It is not that hard, but I want you to think for a while too.

0) By considering $|f_k|^p$ the problem is transformed into a problem on $L^1$.

1) Would Lebesgue domination do? (Why or why not?)

2) Would Monotone convergence do it for us? (Why or why not?)

3) Would Fatou's lemma work? (Why or why not?)

If none of these would work, you might be deeper. (Now I say no more...)