How is this proof for the scalar product rule of limits valid? If we let $K=\lim\limits_{x \to a} f(x),$
and let $c$ be a constant,
Then in order to show that $\lim\limits_{x \to a} cf(x) = cK$,
we must show that there is an $\epsilon$ for every $\delta$ such that
$\lvert cf(x)-cK \rvert < \epsilon$ whenever $\lvert x-a \rvert < \delta$.
The proof claims to prove this by staing:
$$\lvert cf(x)-cK \rvert = \lvert c \rvert \lvert f(x)-K \rvert < \lvert c \rvert  \frac{\epsilon}{\lvert c \rvert} = \epsilon.$$
However, I don't see how this proves anything other than basic manipulation of terms, and I don't see how it relates $\epsilon$ to $\delta$ in any way.

The step missing is saying "So take $\delta>0$ such that $|f(x)-K|<\epsilon/|c|$ whenever $|x-a|<\delta$. Thus, that $\epsilon$ and $\delta$ will work to show that $|cf(x)-cK|<\epsilon$ whenever $|x-a|<\delta$."
Most of the time, this step is left out of proof involving limits since it has the same flavor and is largely the same every time.
One other thing worth mentioning is that this proof doesn't work when $c=0$, but that exceptional case is rather trivial since it just leads to the claim that $\lim_{x \to a}{0}=0$.
