I am trying to check whether the implication $\forall p>1\quad f\in L_p(X,\mu)\Rightarrow f\in L_1(X,\mu)$ is true when $\mu(X)<\infty$. By $L_p(X,\mu)$ I mean the space of Lebesgue integrable functions $f:X\to\mathbb{C}$ with respect to measure $\mu$.
That holds for $p=2$ thanks to the inequality $|f(x)|\leq\frac{1}{2}(|f(x)|^2+1)|$ and the fact that constant functions are Lebesgue-integrable on a domain of finite measure. Does it hold for any $p>1$? If it does, how can it be proved? $\infty$ thanks!
To specify the details of where this question has come to my mind, I add that I am trying to prove the completeness of normed space $L_p(X,\mu)$, where functions are identified as a class of equivalence when they are equal almost everywhere and $\|f\|:=(\int_X |f|^p d\mu)^{1/p}$, by trying to adapt this proof for the case $p=2$.
Hint: use the Hölder's inequality with $f$ and $x\to 1$.