Is it true that a sequence $ f_n \to f $ of measurable functions is bounded by a norm of $ L^p $ then $ f_n $ converges to $f$ in $ L^p $?

Question

Is it true that a sequence $ f_n \to f $ of measurable functions is bounded by a norm of $ L^p $ then $ f_n $ converges to f in $ L^p $?

Is this true? If so, prove, if not, a counter example.

I just know that $ || f_n ||_{p} <M $ and that $\forall$ $\epsilon>0$ $\exists N_0 $ such that $\forall n>N_0 \Rightarrow |f_n -f |< \epsilon $

How to use this to show that

$\lim_{n \to \infty} ||f_n - f ||_{p} =0 $

Answer 1

If we furthermore assume that $\lVert f_n\rVert_p\to \lVert f\rVert_p$, then it is true.

Otherwise, we may be in trouble, for example $f_n(x)=n^{1/p}\mathbf 1_{(0,1/n)}(x)$ where we endow $(0,1)$ with the Borel $\sigma$-algebra and Lebesgue measure.