Convergence of $L^p$ norm to $L^1$ norm as $p\to 1$

Question

Consider $(f_n)_{n\in \mathbb{N}}$ a sequence of real functions which can be uniformly bounded in any $L^p(\mathbb{R}^3)$ for $p>1$ (by uniformly I mean uniformly in $n$ and $p$ ). So we can extract a subsequence which converges weakly to some $f$ in any $L^p(\mathbb{R}^3)$, $p>1$. I wish for $f\in L^1(\mathbb{R}^3)$. It is well known that $$ \lim_{p\to \infty}\| f\|_{L^p(\mathbb{R}^3)}= \|f\|_{L^{\infty}(\mathbb{R}^3)}.$$ In my case I would like to use $$ \lim_{p\to 1}\| f\|_{L^p(\mathbb{R}^3)}= \|f\|_{L^{1}(\mathbb{R}^3)}.$$ I had in mind that this is true but after thinking about it again I have the feeling that I would need $f\in L^1(\mathbb{R}^3)$. Trying to use Hölder as in the case for $p=\infty$ didn't seem to work. I would be grateful for hints whether this limit holds true and also for some hints how to prove it.