Relation between convergences in $L^{p}$ for probability spaces.

Question

I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$.

I'm not sure if $\|f\|_{2} \le \|f\|_{1}$ for any measurable $f$. Anyway, given a sequence $(f_{n})_{n}$ of functions (not necesarily in $L^{p}$), if $(f_{n})_{n} \to f$ in the $L^{2}$-norm, does this imply $(f_{n})_{n} \to f$ in the $L^{1}$-norm? Thank you.