Assume that if $\lim\limits_{x \rightarrow p}f(x) = L$ and $\lim\limits_{x \rightarrow p}g(x) = K$, then
$\lim\limits_{x \rightarrow p}[f(x)+g(x)] = \lim\limits_{x \rightarrow p}f(x) + \lim\limits_{x \rightarrow p}g(x) = L+K$
and
$\lim\limits_{x \rightarrow p}[f(x)g(x)] = \lim\limits_{x \rightarrow p}f(x) \lim\limits_{x \rightarrow p}g(x) = LK$
Prove, by induction, that if $\lim\limits_{x \rightarrow p}f_1(x) = L_1$,$\lim\limits_{x \rightarrow p}f_2(x) = L_2$,...,$\lim\limits_{x \rightarrow p}f_n(x) = L_n$, then, for all $n \ge 2$
$1.$ $\lim\limits_{x \rightarrow p}[f_1(x)+f_2(x)+...+f_n(x)] = L_1+L_2+...+L_n$
$2.$ $\lim\limits_{x \rightarrow p}[f_1(x)f_2(x)...f_n(x)] = L_1L_2...L_n$
Proof of $1$:
Base case $n=2$:
It follows from the property
$\lim\limits_{x \rightarrow p}[f_1(x)+f_2(x)] = L_1+L_2$
Assume for $n=k$
$\lim\limits_{x \rightarrow p}[f_1(x)+f_2(x)+...+f_k(x)] = L_1+L_2+...+L_k$
Case $n=k+1$
$\lim\limits_{x \rightarrow p}[f_1(x)+f_2(x)+...+f_k(x)+f_{k+1}(x)] = \\\ \lim\limits_{x \rightarrow p}[(f_1(x)+f_2(x)+...+f_k(x))+f_{k+1}(x)] = \\\ \lim\limits_{x \rightarrow p}[f_1(x)+f_2(x)+...+f_k(x)]+\lim\limits_{x \rightarrow p}f_{k+1}(x) = \\\ (L_1+L_2+...+L_k)+L_{k+1}$
Now the Proof of $2$:
Base case $n=2$:
It follows from the property
$\lim\limits_{x \rightarrow p}[f_1(x)f_2(x)] = L_1L_2$
Assume for $n=k$
$\lim\limits_{x \rightarrow p}[f_1(x)f_2(x)...f_k(x)] = L_1L_2...L_k$
Case $n=k+1$
$\lim\limits_{x \rightarrow p}[f_1(x)f_2(x)...f_k(x)f_{k+1}(x)] = \\\ \lim\limits_{x \rightarrow p}[(f_1(x)f_2(x)...f_k(x))f_{k+1}(x)] = \\\ \lim\limits_{x \rightarrow p}[f_1(x)f_2(x)...f_k(x)]\lim\limits_{x \rightarrow p}f_{k+1}(x) = \\\ (L_1L_2...L_k)L_{k+1}$
I'm afraid it's not supposed to be this simple, or is it ?