I've taken all basic Calculus courses and Advanced Calculus I (not that I learned any additional information on proving the basic limit laws in that class...), but I still do not know if what is in my textbook (or what I have researched on the net) is the easiest (intuitive) way to prove the basic limit laws (sum , product, reciprocal, and constant).

I came up with arguments that I highly doubt are valid proofs, since I have never seen an explanation this simple in a proof of these laws by those who are much more mathematically sound than I am.

The main theme of the "arguments" for each of the basic limit laws is the following statement.

Since

is an arbitrary number, then

is an arbitrary number

Now before I show the arguments, I make note of the formal definition of the (two sided) limit.

I evaluated the absolute value to instead have:

Since this adjusted formal definition shows that the definition can be broken into two parts (we have two statements because of the or operator), we can just handle the first part

, and we can understand that the argument is similar if we were to tackle the second part of the definition.

Here are the arguments/"proofs" using this theme.  To save space, let (*) represent the statement in the first quote (epsilon times a number is an arbitrary number/another epsilon..., etc.)

Sum Law

Argument

Constant Law

Argument

Product Law

Argument

Reciprocal Law

Argument

We can of course prove the difference law with the results of the sum and constant laws, and we can prove the quotient law using the results of the product and reciprocal laws.

Is (*) (the statement in the first quote) a correct assumption?  If so, then are these arguments valid proofs?