Answer
See explanations.
Work Step by Step
Step 1. Assume $\lim_{x\to c}f(x)=L$. Letting $g(x)=kf(x)$, we can rewrite the limit to be proved as $\lim_{x\to c}g(x)= k\lim_{x\to c}f(x)=kL$. We will consider the case that $k\gt0$.
Step 2. To prove the limit, for any small value $\epsilon\gt0$, we need to find a corresponding $\delta\gt0$ so that for all $x$ in the interval of $|x-c|\lt\delta$, we have $|g(x)-kL|\lt\epsilon$,
Step 3. The last inequality can be rewritten as $-\epsilon\lt g(x)-kL\lt \epsilon$ or $kL-\epsilon\lt kf(x)\lt \epsilon+kL$ which gives $L-\epsilon/k\lt f(x)\lt \epsilon/k+L$ or $-\epsilon/k\lt f(x)-L\lt \epsilon/k$
Step 4. We can choose $\delta=\epsilon/k$ and work the steps backwards to reach the conclusion that for all $x$ in the interval of $|x-c|\lt\delta$, we have $|g(x)-kL|\lt\epsilon$, which proves the limit statement for the Constant Multiple Rule.
Step 5. For the case that $k\lt0$, we can prove the limit in a similar way.
