Confirm limit of this sequence by using definition of limit

[s3a]

Homework Statement

I'm referring to the question and solution for part (b) in the attached TheProblemAndSolution.jpg file.

Homework Equations

Definition of limit.

The Attempt at a Solution

Should the equation with the two things in brackets have absolute value bars instead of brackets?

Also, according to this Wikipedia link ( http://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_Definition ), N should be a natural number and ϵ should be a real number but the solution in my book gives N as a function of ϵ, which means that it could be the case that N is also real and non-natural (such as an irrational number for example). Is there a contradiction here, or is there something I'm missing?

Any input would be GREATLY appreciated!

 

[mfb]

Should the equation with the two things in brackets have absolute value bars instead of brackets?

Yes.

Also, according to this Wikipedia link ( http://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_Definition ), N should be a natural number and ϵ should be a real number but the solution in my book gives N as a function of ϵ, which means that it could be the case that N is also real and non-natural (such as an irrational number for example). Is there a contradiction here, or is there something I'm missing?

You can simply round up. If something is true for all n larger than 245.3, then it is also true for all n larger than 246.

 

[s3a]

Yes.

Thanks for confirming.

You can simply round up. If something is true for all n larger than 245.3, then it is also true for all n larger than 246.

Thanks for this answer too, but there is a bit more I need to understand.

Is it more strict than neccesary (for that Wikipedia link I gave) to require that n and N be natural numbers?

Shouldn't N be allowed to be any real number, and shouldn't n be allowed to be any integer (including negative ones)?

 

[s3a]

Sorry, I double-posted.

 

[s3a]

Actually, I have a feeling it just doesn't matter if N is a natural number or not, but I wanted to know if there are varying definitions (among different textbooks and websites, etc), or if every or most sources have the definition that the Wikipedia link uses.

 

[s3a]

Okay, so, if anyone else cares, I got the answer to my problem.:

Since there is no regulatory body for standardizing definitions in mathematics, not all definitions of the limit of a sequence are exactly the same, but they are all largely similar, so in the case of the problem in this thread, I could use the definition where N is a real number (instead of rounding up to the nearest natural number).

Also, apparently, indices for terms of sequences are always natural numbers.

Also, thanks, mfb.

 

[Fredrik]

There's nothing wrong with a definition that says that N is a positive real number, but it's a bit weird. What the standard definition of the limit of a sequence tells you is this: A real number x is a limit of a sequence S if and only if every open interval centered at x contains all but a finite number of terms of the sequence. The $\varepsilon$-N statement of the definition is just a way to rephrase that in a way that makes it perfectly clear what "all but a finite number" means. So when you want to prove that $x_n\to x$, the idea is to identify a specific term $x_N$ such that it, and all subsequent terms, are in the given interval.

N must depend on $\varepsilon$, because $\varepsilon$ tells you the size of the interval. For example, consider the sequence 1/n and an open interval centered at 0. The smaller the interval, the further you have to go into the sequence to find a term such that all subsequent terms are also in that interval.

Yes, the indices are always integers, in the sense that if they're not, we would call it a "net" instead of a "sequence". A sequence can be defined as a function from the set of natural numbers into some other set. A net is defined as a function from a directed set into some other set. You don't have to learn about nets or directed sets in this course. I just thought it might be interesting to know.