Showing a convergence in $ L^p$

Question

If $ f_n \in L^p(X,m) $ and $f_n \to 0 $ a.e with respect to $m$ and $||f_n||_p \to ||f||_p $ as $n \to \infty$ then can we say that for all $r \in [1,\infty] $ we have $$ ||f_n-f||_r \to 0 $$

Can someone help me , a little hint how to use the assumptions. Thanks

Answer 1

The claim is in general not correct. Consider for example $X := [1,\infty)$ endowed with the Lebesgue measure and

$$f_n(x) := \frac{1}{n} \frac{1}{x}.$$

Obviously, $f_n \to f:=0$ almost everywhere, $\|f_n\|_{L^2} \to 0$, but for $r=1$

$$\|f_n-f\|_{L^1} = \|f_n\|_{L^1} = \infty.$$