# Constructing a dominating integrable function for a sequence of integrable functions $f_n \to f$?

I have a sequence of functions $$f_n \to f$$. I have shown that both $$f_n$$ and $$f$$ are integrable. In order to complete my proof, I needed to apply the dominated convergence theorem to yield

$$\int f_n d\mu \to \int f d\mu$$

Fortunately for me, since I was told $$f$$ was integrable, and I constructed $$f_n$$ so that $$|f_n|\leq |f|$$, I could simply take the dominating function as $$f$$.

My question however is what if we were not told $$f$$ was integrable. What if we only knew:

1. Each $$f_n$$ itself is integrable

2. $$f_n \to f$$

Can you always construct a dominating integrable function then? Clearly if only $$(1)$$ then holds plenty of counter examples exist. But I am curious if the fact $$f_n$$ converges to $$f$$ ensures such a dominating integrable function exists.