Convergence with indicator functions

Solution 1:

Pointwise convergence means you look at some fixed $x\in(0,\infty)$ and you ask whether $f_n(x)\to 0$. You do this for each and every $x$ in your domain separately.

In this example it is certainly true that $f_n\to 0$ everywhere, because for any fixed $x\in(0,\infty)$ there's $n_0\in\mathbb{N}$ large enough such that $f_n(x)=0$ for all $n\geq n_0$. You say "If n is large enough, in such a way that $x\notin [n,n+1]$..." but this sentence is meaningless: which $x$ are you referring to? For any fixed $x$ we can find $n$ that is large enough, but we can't find $n$ that is large enough for all $x$.

(Since $f_n\to 0$ pointwise then in particular $f_n\to 0$ a.e. -- after all, it converges everywhere).

Regarding $L_\infty$, this is simply because the norm of each and every element of the sequence $f_n$ is $1$. Therefore you get a constant sequence which certainly converges to $1$.