I am new to limits. I wanted to prove the product law, however, I am not confident since it's different from other sources. Please verify. Any suggestions are welcomed!
Product law:
If $\lim_{x \to c} f(x) = a$ and $\lim_{x \to c} g(x) = b$, then $\lim_{x \to c} f(x)g(x) = ab$
Proof:
$|x-c| < \delta_1 \implies |f(x)-a| < \epsilon_1$
$|x-c| < \delta_2 \implies |g(x)-b| < \epsilon_2$
Since $|g(x)-b| < \epsilon_2$, we have $-\epsilon_2 < g(x)-b < \epsilon_2 \implies b-\epsilon_2<g(x)<b+\epsilon_2 \implies |g(x)| < max \{ |b-\epsilon_2|, |b+\epsilon_2| \}$
$|f(x)g(x)-ab| = |g(x)(f(x)-a)+a(g(x)-b)| \leq |g(x)|\epsilon_1+|a|\epsilon_2 < max \{ |b-\epsilon_2|, |b+\epsilon_2| \}\epsilon_1+|a|\epsilon_2$
Which means $|x-c| < min \{ \delta_1, \delta_2 \} \implies |f(x)g(x)-ab| < max \{ |b-\epsilon_2|, |b+\epsilon_2| \}\epsilon_1+|a|\epsilon_2$