Understanding an example on pg.83 (Royden $4^{th}$ edition).

Question

The example is given below:

My questions are:

1- I can see that $f_{1} = 1$ when $x \in (0,1),$ $f_{2} = 2$ when $x \in (0,1/2),$ $f_{3} = 3$ when $x \in (0,1/3),$ $f_{4} = 4$ when $x \in (0,1/4)....$

So I do not understand how $\{f_{n}\} \rightarrow f = 0$ on $E,$ could anyone explains this for me please?

2- why we are removing the zero from $E$ and $\chi$ domain and why are we removing $1/n$ from $\chi$?

Answer 1

Let $x_0 \in (0,1]$

Then exists $m \in\ \Bbb{N}$ sicu that $\frac{1}{n}<x_0,\forall n \geq m$

Thus $x_0 \notin (0,\frac{1}{n}],\forall n \geq m\Longrightarrow f_n(x_0)=0,\forall n \geq m$

Thus $f_n(x_0) \to 0$

We remove the zero from the intervals because then we would have that $f_n(0)=1,\forall n \in \Bbb{N}$

But again it does not matter since the integrals over singletons are zero.