pointwise convergence on interval and in $ L^p $

Question

I need to prove that sequence of functions $ (f_n)_{n=1}^{\infty} $is pointwise convergent on $[0,1]$ but it is not convergent in the space $ L_2[0,1] $

If I showed that $ f_n \to 0 $ on $ [0,1] $ is it enough to show that $ ||f_n-0||_{L_2[0,1]} \not\to 0 $ ?

Is it always true that if $ f_n \to f $ pointwise then $ ||f_n-f|| \to 0 $ ?

Answer 1

If $f_n=f$ for all $n$ then $f_n \to f$ pointwise and $\|f_n-f\|$ also tends to $0$. So you cannot prove that $\|f_n-f\|$ does not tend to $0$. You can only given an example where we have $f_n \to f$ pointwise but $\|f_n-f\|$ does not tend to $0$. Such an example is given by $f_n=\sqrt n I_{(0,\frac 1 n)}$.