Exercise chapter 2.12 real analysis by E.M. Stein and R.Shakarchi 
Exercise chapter 2.12 real analysis by E.M. Stein and R.Shakarchi :
Show that there are $f \in L^1(\mathbb{R^d})$ and a sequence $\{ f_n \}$with $f_n  \in L^1(\mathbb{R^d})$ such
that
$\|f_n-f\|_1 \to 0$
but $f_n \to f$ for no $x$ (pointwise convergence).
[Hint: In R, let $f_n:=\chi_ {I_n}$, where $ I_n$ is an appropriately chosen sequence of intervals
with $m(I_n) \to 0$.]

I find :
$$\chi_{[0,1]}, \chi_{[0,1/2]}, \chi_{[1/2,1]},\chi_{[0,1/3]},\chi_{[1/3,2/3]},\chi_{[2/3,1]}, \dots $$
This sequence $\to 0$ in $L^1,$ but is  pointwise nowhere ? Why? Is this true: for any given $x$ there are many infinitly $n$ shuch that $f_n (x)=1$?
My example is for $d=1$ or $ L^1(\mathbb{R})$. How we can show for all $d \in\mathbb{N} $ or $L^1(\mathbb{R^d})$  it is true ?
 A: The idea can be adapted: instead of covering the unit interval by intervals of smaller and smaller length, we cover the unit cube by cubes of smaller and smaller size. More precisely, define 
$$
C_{n,\mathbf{k}}=:\prod_{i=1}^d\left[\frac{k_i-1}{2^n},\frac{k_i}{2^n}\right], \quad 1\leqslant k_i\leqslant 2^n,1\leqslant i\leqslant d,n\geqslant 1.$$
Let $N_n:=\sum_{j=1}^n2^{jd}$ and for a fixed $n$, label the family $\left(C_{n,\mathbf{k}}\right)_{1\leqslant k_i\leqslant 2^n,1\leqslant i\leqslant d}$ as $\left(B_j\right)_{j=N_{n-1}+1}^{N_n}$ (the order does not matter). Then define $f_j$ as the indicator of $B_j$. In this way, the sequence $\left(\left\lVert f_j\right\rVert_1\right)_{j\geqslant 1}$ converges to $0$ but for any $x\in \left[0,1\right]^d$, the sequence $\left(f_j\left(x\right)\right)_{j\geqslant 1}$ does not converge to $0$. In order to have this for any $x\in\mathbb R^d$, define 
$$
g_j\colon x\mapsto \sum_{\mathbf{k}\in \mathbb Z^d} 2^{-\sum_{i=1}^d\left\lvert k_i\right\rvert} f_j\left(x+\mathbf{k}\right).
$$
