I am currently relearning calculus after ages of not having much to do with it, and I'm looking at the proofs for the basic limit laws. I was wondering if there is a "simpler"/ more elegant way of proving them, as opposed to the standard Proofs that use the epsilon-delta definition. The reason I ask is because I noticed the following:
If we let the limit Process be represented by "@", such that the limit of f(X) as X approaches n becomes "n @ f(X)" then the limit laws just become Distributive Property statements.
For instance, the addition law becomes: n @ [ f(X) + g(X) ] = n @ f(X) + n @ g(X)
This leads me to think that there is some way of proving the limit laws using symmetry or set theory or somesuch. Am I correct in this? The way I figure it such a proof might go as follows:
Prove that the limit operation belongs to a certain kind of operations
Prove that this kind of operations is distributive with respect to the 4 common algebraic operations.
This is blowing my mind. Is there such a proof?
Just some thoughts on limits.
(1).One difficulty is that $f(x)+g(x)$, or $f(x)g(x)$ may have a limit as $x\to 0$ even though $f(x)$ and $g(x)$ do not have limits as $x\to 0.$ That is,it is true that if $a=\lim f(x)$ and $b=\lim g(x)$ (as $x\to 0$) then $f(x)+g(x)\to a+b$ and $f(x)g(x)\to a b$, but the reverse implication does not hold. So the distributive property is "one-directional."
(2).It is often easier to use the following equivalent def'n of $\lim_{n\to \infty}x_n=y$ : That $y$ is the unique point such that for every neighborhood $U$ of $y$, the set $\{n: x_n\not \in U\}$ is finite.
(3).A set-theoretic topologic equivalent of $\lim_{n\to \infty}x_n=y$ is $\{y\}=\cap_{n\in N}$ Cl$(\{x_m:m\geq n\}).$
(4).You may see $T=\lim \sup_{n\to \infty}S$ when $S=(S_n)_{n\in N}$ is a sequence of sets. $T$ is the set of all,and, only ,those $x$ such that $\{n : x\in S_n\}$ is infinite. That is,$T=\cap_{n\in N}\cup_{m\geq n}S_m.$