Topic: δ-ε proofs

I am to write a plain-text essay on delta-epsilon; proofs for Calculus I, as some sort of cross-disciplinary testing my university is doing. Or whatever. The point is that I realized my knowledge of the Cauchy criterion is quite limited and has come to a halt due to a probably stupid misunderstanding! So here is what I have in outline. My gap in understanding is how the relationship between d [=delta] and e [=epsilon] is established. Also, to prove in the general case; I want to have command of the proof, because things get very muddy when dealing with complex transcendental functions and their δ-ε proofs. So I am talking it through and would appreciate some help. Besides, it's math. And everybody likes math.


Ae>0 Ed>0 | o<|x-a|<d --> |f(x)-L|<e

where [lim x->a] f(x) = L

So my d value must be substantially close such that no critical points exist between |x-a| which could disrupt the continuity of f(x).

So my corresponding e is going to be some value defined by the criterion of d = (a-d, a)U(a, a+d) which will create corresponding points if f(x) is continuous at (L-e, e)U(e, L+e).

The last part of the argument states:

0<|x-a|<d --> |?(x)-L|<e

which means to me, and this is my point of [b]confusion[/i], that I want to define d according to the above restrictions and discover if a corresponding e exists. So following the rules of implication, the only way this statement would be true and the limit false is if 0<|x-a|<d is true and |?(x)-L|<e is false.

So how do I establish a relationship between d and e in the form of an equation, or do I need to do this?

I know what I want to do, I just don't know how I need to do it.

[EDIT: GRR. How do I represent mathematical symbols?]

[EDIT: there we go.]

(Edited by cfb on 04-11-2007 00:41)

quote:

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GRR. How do I represent mathematical symbols?

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Using MATH ML, or by their name or in last resort using images.

I can bring a proof only to this point:

lim [x->a] f(x) = L

|x-a|<d --> |f(x)-L|<e

So for an easy example

lim [x->2] 4x + 2 = 10

|x-2|<d --> |4x+2 - 10|<e
|x-2|<d --> |4x-8|<e = |4(x-2)|<e
|x-2|<d --> |x-2|<(e/4)

However I'm not sure how to proceed from the definitions I provided above - how is the relationship established from here? I know what to do from the textbook - i.e. the process - but not when I try to reconstruct it. I've hit the point where when writing a lit paper on Tennyson or whatnot, one cannot think of any more, and the thesaurus has run out, of synonyms for "longing" and "craft." You just repeat the same shit over and over again.....ha.

(Edited by cfb on 04-11-2007 00:58)