I need to prove that a certain function is continuous (with an epsilon-delta proof). I want to restrict my $\delta$ to be $\delta \leq \frac{1}{2}$ (bigger than $\frac{1}{2}$ complicates things). If I were to choose $\delta = \epsilon$ I believe my proof wouldn't be valid for all $\epsilon$ since I'm restricting $\delta$ and thereby $\epsilon$ to be smaller than or equal to $\frac{1}{2}$.
How can I solve this?
Can I just say "choose $\delta \leq \epsilon$" or do I really need an equal sign in between those two?
If so, could I solve this by saying "choose $\delta = \min\{\frac{1}{2},\epsilon\}$? In this case, do I have to write out two proofs? One for $\delta = \frac{1}{2}$ and one for $\delta = \epsilon$?