I am trying to work through an exercise (on my own) out of Resnick's A Probability Path.
One question states the following:
Suppose $-\infty and assume we have an indicator function of the form $1_{(a,b]}(x)$. Can this function be approximated by bounded and continuous functions.
Said another way, does there exist a sequence $f_n: 0\le f_n\le 1$ such that $f_n\rightarrow1_{(a,b]}$ pointwise.
Using the hint in the text, my attempted solution was:
Attempt at a solution:
$f_n$ such that: $f_n(x) = \begin{cases} 0, & x \leq a, x \geq b + \frac1n \\ 1, & a + \frac1n < x \leq b \end{cases}$ and linear otherwise
The problem (or one problem?) is that this doesnt appear continuous to me. I could also use some help in learning how to be more explicit about limits. For example, why does $x>b$ imply $f_n\rightarrow 0$. I "know" this intuitively, but obviously need help explaining it with precision.
Thanks as always!