Regarding epsilon proofs with N1,N2,Max,Min

[Unassuming]

This is a famous proof that utilizes a common notion.

Theorem. Limits are unique.

let n>N_1 such that blah blah blah is less than epsilon over 2,

let n>N_2 such that blah blah blah is less than epsilon over 2.

For n> max{N_1,N_2},

blah blah blah < blah = epsilon.

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Somebody presented a question today to my professor of why we can't say in the first place,

let n > N such that blah blah blah is less than epsilon over 2, AND!

also that blah blah blah is less than epsilon over 2.

Then we skip the max stuff and finish up.

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My professor said this was "ok" and we proceeded. My question is, is this "ok"? and why is it not presented in book then?

 

[Unassuming]

I guess I need to be more rigorous here. I will clean it up later today when I get to school hopefully. Sorry.

 

[arildno]

As long as blah-blahiness represents, for you, the summit of precision, don't bother.

 

[Unassuming]

Well, the problem with my writing was that I was trying to generalize the question to many of these types of proofs. I actually thought at the time that it was a good idea and if I would have read something similar, I think I might have understood what the amateur analyst (me) was trying to convey. Many people in my class were wondering the same thing.

My point is that I have seen many proofs involving two N's (N_1 and N_2), and sometimes more. Then we take the max ( or in some cases the min ) of all these N's. Why can't we, like the student asked, take one N from the beginning as long as we state the requirements of that N.

When I taught high school the students asked some very silly (i.e. blah blah blah) questions. I felt that I could understand what they were getting at though because I tried to understand (after I got off my high horse).

 

[tiny-tim]

My point is that I have seen many proofs involving two N's (N_1 and N_2), and sometimes more. Then we take the max ( or in some cases the min ) of all these N's. Why can't we, like the student asked, take one N from the beginning as long as we state the requirements of that N.

Hi Unassuming!

You can do it that way, but it's less rigorous …

if you start from the definition of limit, you will get two Ns, not one …

professors and books usually prefer proofs that are 100% rigorous.

 

[HallsofIvy]

You can, as your professor agreed, say "There exist N such that if n> N,... both are true".

But that doesnt' follow immediately from what you were given. From one limit you know "there exist N1 such that if n> N1...". From the other, "there exist N2 such that if n> N2...". You can THEN say "let N= larger of N1 and N2" so that n> N implies both n> N1 and n> N2.

By the way, try to avoid "BLAH,BLAH,BLAH". That implies something not worth listening to and if you really feel that way, you shouldn't be taking the course! Better to use "...": simpler and less confrontational.

 

[Unassuming]

Ok. So it doesn't directly follow from the definition. I figured it was something, but I like writing the max{N_1,N_2} part anyway. It feels empowering.

Sorry about the blah. I like analysis and I didn't mean to sound uninterested!

Thanks for the answers.