This is a four-part question I stumbled upon. I have attempted the first two, and believe them to be mostly correct, but the last two have me stumped.
Consider the normed space $(X,||\cdot||)=(C(a,b), ||\cdot||_{L^1})$
$\forall n\geq1$, let $b_d,b_{d-1},...b_0$ be the binary expansion of $n$ with $b_i\in \{0,1\}$ for $1<i\leq d$ ,and $b_d=1$
Let $k=n-2^d=b_{d-1}2^{d-1}+...+b_{d}$. For $0\leq k\leq 2^{d-1}$, define the following two sequences of functions:
$f_n,g_n:(0,1)\rightarrow \mathbb{R}$ with $f_n(x)=n-n^2x \chi_{(0,1/n]}$
$g_n(x)=1-2^d|x-k2^d|$ if $|x-k2^d|<2^{-d}$ on $(0,1)$
Questions
$(1)$ Compute $lim_{n\rightarrow \infty}f_n(x)=h(x)$
Attempt: of course, for $x=0$, we have just have $f_n(0)=n\chi_{(0,1/n]}$ which just converges to $h(x)=n\chi_{\{0\}}$, since this is a constant sequence of functions on a shrinking interval.
For $x=1$, we have $f_n(1)=n-n^2x\chi_{(0,1/n]}$, which converges to $h(x)=n-n^2\chi_{\{0\}}$.
From these two cases, we see that the the convergent is pointwise since we have a discontinuous function as a limit.
for $0<x<1$, I believe we should just get the same as with the $x=1$ case although I am not clear on how to prove it rigorously.
$(2)$
Does $f_n$ converge to $h$ in the $L^1$ norm?
Attempt:
we NTS $lim_{n\rightarrow \infty}||f_n-h||_{L^1}=0$.
Assuming I found the limit function correctly in part $(1)$, we have that $h(x)=n\chi_{\{0\}}+(n-n^2)\chi_{(0,1]}$
so we have $\lim_{n\rightarrow \infty}\int_{0}^{1}|f_n-h|dx$
Here is where I get stuck. I know for $f=g$ a.e. the integrals of the two functions are the same, but I don't know if we are justified doing that when taking limits of sequences of functions. I also don't know if there are different rules for pointwise vs. uniform convergence towards the limit functions.
Assuming I am justified in removing the case $f_n(0)$ the integral becomes $lim_{n\rightarrow \infty} \int_{0}^{1/n}|(n-n^2x)-(n-n^2x)|dx$ and this certainly converges in the $L^1$ norm.
$(3)$ compute $||g_n||_{L^1}$
I honestly am not seeing how to do this.
$(4)$
Find a dense subset $S\subset [0,1]$ so that for each $x\in S$, we have a subsequence $(n_j$ where $lim_{j\rightarrow \infty}g_{n_{j}}=1$ and $lim_{j\rightarrow \infty}g_{n_{j+1}}=0$. We want to argue then the limit doesn't exist for any point $x\in S$.
I tried to do something with Arzela-Ascoli but had no success.