If $f_n\to f$ in $L^1$ can we derive that the functions $f_n$ are bounded by an integrable function?

Question

Let $f_n,f$ be positive functions such that $f\in L^1(\Omega)$ and $f_n\in L^p(\Omega)\,\,\forall\,1\leq p<\infty.$ If $f_n\to f$ in $L^1$ can we derive that the functions $f_n$ are bounded by an integrable function? I was thinking about writing $$f_n\leq |\,f_n-f\,|+f.$$ I would appreciate any idea. thank you.