I am reading on a proof of the product law for limits and one of the steps confuses me. I'll just show the important inequality and omit the detials
$|f(x)g(x) - f(a)g(a)| \leq |f(x)||g(x) - g(a)| + |g(a)||f(x) - f(a)| < |1 + f(a)|\dfrac{\epsilon}{2(|f(a)| + 1)} + |g(a)|\dfrac{\epsilon}{2(|g(a)|+1)} = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon $
Where we assumed f and g have limits and that $|f(x) - f(a)| < \min \{1,\dfrac{\epsilon}{2(|g(a)| + 1)}\}$ and $|g(x) - g(a)|<\dfrac{\epsilon}{2(|f(a)|+1)}$
How on earth did they come up with those $\epsilon$? And how did $|g(a)|$ cancel out with $2(|g(a)|+1)$ in the bottom to give that 1/2?
I found this proof in Spivak's book. Is it a typo? Maybe it should be < instead of =?