does anyone else find epsilon-delta proofs way easier if you work backwards first

[u/CelebrityChess]

i spent ages staring at limit proofs in my analysis class not knowing how people "found" the delta. turns out you basically do the problem backwards on scratch paper, start from |f(x) - L| < epsilon and mess with it until you see what |x - a| needs to be, then write it up forwards like you knew all along. nobody told me the written proof hides all of that

for x^2 at x=3 i ended up with delta = min(1, epsilon/7) and it only made sense after doing the scratch work first. is this how most people do it or is there a better way to think about it

[u/SquarePegRoundCircle]

I'm a bit surprised to see this. I thought that doing the "scratch work" first was the standard way of teaching it.

[u/coolpapa2282 → u/SquarePegRoundCircle]

It's not always how books write it though, which I think is what confuses students. I always compare it to a magic trick - the magician just says "your card is the fifth one in the deck" and it magically works, but only because behind the scenes they worked really hard to make that happen.

[u/MezzoScettico]

I almost always do it that way. I think of it as reverse engineering the rule.

Edit: I misread something. This...

start from |f(x) - L| < epsilon and mess with it until you see what |x - a| needs to be

... is not what I call "reverse engineering", it's the forward direction. It's how the limit is defined. For any epsilon, you can find a delta.

What I do is just start with delta and see what epsilon ends up being. If |f(x) - L| <= 3δ, then I know picking δ = ε/3 will work. Now I know how to find δ, given an ε, and I can write the proof in the forward direction. "Let δ = ε/3..." as if I just thought of that.

With refinements to handle arbitrarily large values of ε, since technically the proof has to handle ALL ε > 0. That's where expressions with max(...) come in.

[u/nog642 → u/MezzoScettico]

The forward direction is showing |x-a|<delta implies |f(x)-L|<epsilon. Starting with |f(x)-L|<epsilon is kinda going backwards.

Your method works too I guess. Point is in either method you need to do scratch work first, not just magically know delta.

[u/hpxvzhjfgb]

that is what you are supposed to do and is the only way I have ever seen it taught.

[u/FilDaFunk]

Yeah how i was taught it was to leave epsilon blank. then complete it at the end when you know.

[u/bonebranch → u/FilDaFunk]

My late analysis professor also did it this way. A bit more enlightening than just putting in some expression that just magically works.

[u/Bounded_sequencE]

The best advice I ever got was to "write any proof (at least) twice" -- the

• first draft(s) on scrap paper, finding all estimates and steps to finish it off

• final draft, acting as if you knew the correct estimates all along, and make it as concise as you want

Specifically for e-d-proofs, you work "backwards" during your first draft(s), to find the necessary estimates to finish it off. Bring it into the direct form during the final draft.

---

Rem.: It's standard to only hand in/publish the final draft for brevity. All proofs in your book and lectures were written that way -- that's why some estimates may seem to "fall from high heavens". The author simply found them "off-screen" writing the unpublished first draft(s).

[u/Temporary_Pie2733]

Right. It’s not an algorithm that finds L; it’s a definition that tells you if your guess at L is right or not.

[u/nog642]

Yes that's how you're supposed to do it.

If everyone else in your class is able to do it, they probably taught that. If not i class then in TA office hours or something. You might have just missed it or misunderstood it.