For example, let us say I have used an $\epsilon$-$\delta$ proof to show that $\lim\limits_{x \to a} f(x) = L$.
Then to prove all of the other properties of a limit, can't I just continuously substitute?
For limit of a constant, $f(x)=c$; for sum and different properties $f(x)=g(x)+h(x)$ and $f(x)=g(x)-h(x)$, respectively; for product and quotient properties $f(x)=g(x)h(x)$ and $f(x)=\frac{g(x)}{h(x)}$, respectively; finally, for constant multiple $f(x)=g(x)h(x)$, where $g(x)=c$ and $c$ is some arbitrary constant.
This feels cheaty. I missing something? For example, since I am assuming I have already shown $\lim\limits_{x \to a} f(x) = L$ via an $\epsilon$-$\delta$ proof, does these mean all of these proofs are just $\epsilon$-$\delta$ proofs by proxy? As in, to show $\lim\limits_{x \to a} f(x) = L$ for $f(x)=g(x)+h(x)$, in the $\epsilon$-$\delta$ proof, will I just have to effectively prove $\lim\limits_{x \to a} g(x)+h(x) = L$, thereby making the $f(x)$ obselete?