Suppose $X$ is a measurable space with $\mu$ the measure and $\mu (X) < \infty$ .
If ${f_n}$ be a Cauchy sequence in $L^2(X,F,\mu)$, then is it a Cauchy sequence in $L^1(X,F,\mu)$ too?
I know that since $\mu(X)<\infty$ we have $L^2(X,F,\mu)\subseteq L^1(X,F,\mu) $.
Sure. Suppose $f_{n}$ is Cauchy in $L^{2}$. Then, by Holder's inequality and the fact that $\mu(X)<\infty$,
$$ \|f_{n}-f_{m}\|_{1}\leq \|f_{n}-f_{m}\|_{2}\|1_{X}\|_{2}=\|f_{n}-f_{m}\|_2\mu(X)^{1/2}\to 0\quad\text{ as }n,m\to\infty. $$