>>12033675Technically all the epsilon variations are just to make the proof pretty. It doesn't matter if your quantity is less than epsilon or epsilon over two but Mathematicians of course want elegance and clarity in the final statement.
What you want to do is FIRST just let epsilon, delta, etc. be arbitrary and go through the motions of the proof to see what your final sum is. THEN you can go back and fix your delta or whatnot so that the inequality relates to just epsilon (since a lot of our definitions relate simply to epsilon, e.g. consider the definition of sequence convergence).
If you don't know what I mean, essentially in Elementary Analysis what you want to do is find or design inequalities by relating definitions, etc. so when you check a statement (usually you want to say that it's less than an arbitrary positive number) you can write out an inequality by examining the individual terms and their respective inequalities. Hence the fixing aspect, since often multiple terms will be less than an arbitrary positive number by definition but if you let them all be epsilon, you'll come out with some proportion and in more convoluted proofs, you come out with proportions involving other quantities. E.g. you end up with multiple terms when you apply the triangle inequality several times, etc.
Analysis isn't that strange; how novel it seems when you're first exposed that may make it seem so. Hopefully this doesn't confuse you/ I don't mislead too much, this is a fairly broad suject and of course I'm only thinking about a small portion that I think relates to your frustrations.
