So I decided to go back and study the proofs about limit theorems. This one stumped me;
Theorem: If $\displaystyle\lim_{x\to a} f(x)=L$ and $\displaystyle\lim_{x\to a} g(x)=M$ then $\displaystyle\lim_{x\to a}[f(x)+g(x)]=L+M$.
I wonder if I can link? Anyway, the proof is here: http://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx at the 'proof of 2' note. My question is, why do we have to choose the $\delta$ as the smaller of the two; $\delta _1$ and $\delta_2$? Also, why is it =$\epsilon $ at last part, shouldn't it be