Valid Proof for Properties of Cumulative Distribution Functions? I am currently reading through Mathematical Statistics by Hogg and Craig (8th edition)
One of the questions is to prove a few properties of Cumulative Distribution Functions (CDFs). Here, for a given random variable $X$, I use the notation $F_X(x)$ to represent its CDF. I represent its PDF by $f_X(x)$
The first asks to prove that $$\lim_{x \to -\infty} F_X(x)= 0 \text{ (lower bound of F_X is 0)}$$
The second question asks to prove that $$\lim_{x \to \infty} F_X(x)= 1 \text{ (upper bound of F_X is 0)}$$
I attempt to prove the first by using properties of limits and integrals as well as the definition of $F_x$. They're quite different than the proofs provided by online answers and they seem a little too... simple? So as a person not too familiar with proving things I'm a bit skeptical.
Anyways, my proof for the first one is as follows:
$$\lim_{x \to -\infty} F_x= 0$$
$$\lim_{x \to -\infty} \int_{-\infty}^x f_X(x)dx = 0 $$
$$\lim_{x \to -\infty} \int_{-\infty}^{-\infty} f_X(x)dx = 0 $$
since integral at a point is equal to $0$,
$$\lim_{x \to -\infty} 0 = 0 $$
$$0 = 0 $$
And for the second:
$$\lim_{x \to \infty} F_X(x)= 1$$
$$\lim_{x \to \infty} \int_{-\infty}^x f_X(x)dx = 1 $$
$$\lim_{x \to \infty} \int_{-\infty}^{\infty} f_X(x)dx = 1 $$
$$\text { by definition the PDF integrates to 1 over all X}$$
$$\lim_{x \to \infty} 1 = 1 $$
$$1 = 1 $$
The answers provided on Slader are:


Which involves more logic from set theory. I sort of understanding why this works, but it seems... really unintuitive to me? I don't think I would ever think of that. But I do have a feeling that my proofs are wrong somewhere.
Thanks,
A
 A: There are a few problematic points in terms of mathematical writing in your argument. The first is in the following  identity, aside any matter of keeping different names for the integrand variables,
$$
\lim_{x \to -\infty} \int_{-\infty}^{x} f_X(x)dx
=
\lim_{x \to -\infty} \int_{-\infty}^{-\infty} f_X(x)dx.
$$
This identity suggests that, you are using
$$ 
\lim_{z \to z_0} g(z)=\lim_{z \to z_0} g(z_0),
$$
which is true (for a continuous function) as the right-hand side is a constant, but unnecessary.
Also, deriving identities of the form $0=0$ is unnecessary, notice you could perform the whole argument using the integral, to then say that it is equal to $0$, without carrying the term $=0$ in multiple lines.
But finally, as @Kavi Rama Murthy mentioned in the comments, not every random variable has a well-defined density. Indeed, even for continuous random variables there are counter-examples. So in order to keep generality, you need to write in terms of events/sets and use the continuity property of probability measures themselves.
