I'm struggling to find examples of a sequence of functions in two different cases: let $1<p<2$
- functions $(f)_n \in L^p(\Bbb{R},dx)\cap L^1(\Bbb{R},dx)$ such that $(f)_n$ converges strongly in $L^p(\Bbb{R},dx)$ but not in $L^1(\Bbb{R},dx)$
- functions $(f)_n \in L^p(\Bbb{R},dx)\cap L^2(\Bbb{R},dx)$ such that $(f)_n$ converges strongly in $L^p(\Bbb{R},dx)$ but not in $L^2(\Bbb{R},dx)$
I've tried with a couple of functions involving characteristic functions on sets involving $n$, but can't seem to breakthrough. I appreciate any help!
Edit: for the first sequence I tried $f_n(x) =\frac{1}{n^{1/p}} \Bbb{1}_{[0,n]}(x)$, but the problem is that it's convergent in both $L^p$ and $L^1$ norms. I tried another similar sequence but the problem is always that I can't find a function that doesn't converge in neither norm