How to prove that a sequence $f_{n} \in L^{1}$?

Question

I'm trying to studying Banach and Hilbert spaces.

I'm interested to clearly understand when a sequence of functions $f_{n}$ begin to a Banach space ($L^{1}$) or to a Hilbert space ($L^{2}$).

I think that $f_{n}$ must be measurable and integrable, but I'm asking to you a rigorous method to prove that and why we do in such way.

EDIT: An exemple for $f_n$ is $$f_{n}=\frac{2x^{3}}{n+x^{4}}$$ with $n\ge1$ and $x \in [0+\infty)$.

Thank you very much.