$L^p$ convergence and $L^q$ boundedness implies $L^q$ boundedness.

Question

Suppose $X$ is a probability space, $1\leq p < q < \infty$ and that $(f_n)\subset L^q $. I want to verify the following statement:

If $||f_n||_q \leq M$ for all $n$ and $f_n \rightarrow f$ in $L^p$, then $f\in L^q $ and $||f||_q\leq M$.

I think this is true but cannot come up with any good idea to verify this... Any hint would be really appreciated! Thanks in advance.

Answer 1

Denoting a convergence a.e. subsequence of $(f_{n})$ by itself, we have by Fatou's Lemma that $\|f\|_{L^{q}}\leq\liminf_{n\rightarrow\infty}\|f_{n}\|_{L^{q}}\leq M$.

Answer 2

I solved.

Since $f_n \rightarrow f$ in $L^p$, $\exists f_{n_k}\rightarrow f$ a.e.

Hence, $M^q\geq \liminf \int |f_{n_k}|^q\geq \int|f|^q$.