1. ## Analysis module help

i'm trying to study for an Analysis exam, but im gettin stuck on something which i imagine is quite trivial!!

here is an example from something we were taught in lectures.

lim $\displaystyle \frac{1}{n}$ = 0 (n $\displaystyle \rightarrow$ $\displaystyle \inf$)

and the proof is

$\displaystyle \epsilon$ > 0 given

take N($\displaystyle \epsilon$) s.t. N($\displaystyle \epsilon$) > $\displaystyle \frac{1}{\epsilon}$

for n $\displaystyle \geq$N($\displaystyle \epsilon$) , we have

|$\displaystyle \frac{1}{n} - 0| = \frac{1}{n}$< $\displaystyle \epsilon$

what i'm having trouble with is what $\displaystyle \epsilon$ actually is used for, and consequentially what N($\displaystyle \epsilon$) means.
any help (in the simplest form) would be very much appreciated.

2. Originally Posted by mathmonster
i'm trying to study for an Analysis exam, but im gettin stuck on something which i imagine is quite trivial!!

here is an example from something we were taught in lectures. (E meaning epsilon, and [function] means absolute value of said function, because i dont know how to do absolute value signs)

lim 1/n = 0 (as n goes to infinity)

and the proof is

E>0 given

take N(E) s.t. N(E) > 1/E

for n >= N(E) , we have

[1/n - 0] = 1/n < E

what i'm having trouble with is what E actually is used for, and consequentially what N(E) means.
any help (in the simplest form) would be very much appreciated.
N(E) means a natural number dependent on E.
Consequently, trying to understand it practically is simple.

$\displaystyle 1,\frac12,\frac13,\frac14,.......$ is a sequence given to you.

What this definition essentially claims is:
If given any positive real number(generally we denote it by E) to you, then can you find a tail of the sequence for which all the numbers of the tail are less than that E.

Say I give you $\displaystyle E = \frac{2}{213421}$, my job is to find a tail for your sequence. I say "hey look, I notice that $\displaystyle \frac{2}{213421} \geq 10^{-7} = \frac1{10000000}$. Oh and that is the 10000000th term in your sequence. And from there every term is less than this number. In other words, the tail of your sequence starting from the 10000000th term satisfies the definition."

So one such $\displaystyle N(\frac{2}{213421}) = 10000000$.
Two things noteworthy here:
1) 10000000 is not at all special.
2) And I have proved it for one E, how do we prove this for every E.

For 2, here is the answer.
Notice this line in the proof:

E>0 given
take N(E) s.t. N(E) > 1/E
Why do you think you can do that? If you can tell me this I will continue with my explanation, if you need it.

3. Originally Posted by mathmonster
i'm trying to study for an Analysis exam, but im gettin stuck on something which i imagine is quite trivial!!

here is an example from something we were taught in lectures.

lim $\displaystyle \frac{1}{n}$ = 0 (n $\displaystyle \rightarrow$ $\displaystyle \inf$)

and the proof is

$\displaystyle \epsilon$ > 0 given

take N($\displaystyle \epsilon$) s.t. N($\displaystyle \epsilon$) > $\displaystyle \frac{1}{\epsilon}$

for n $\displaystyle \geq$N($\displaystyle \epsilon$) , we have

|$\displaystyle \frac{1}{n} - 0| = \frac{1}{n}$< $\displaystyle \epsilon$

what i'm having trouble with is what $\displaystyle \epsilon$ actually is used for, and consequentially what N($\displaystyle \epsilon$) means.
any help (in the simplest form) would be very much appreciated.
It means that for any value $\displaystyle \epsilon$> 0 you care to nominate, a value of n can always be found (by taking n > N) such that the difference between 1/n and 0 is less than $\displaystyle \epsilon$.

N will typically be a function of $\displaystyle \epsilon$, that is N = N($\displaystyle \epsilon$).

Study the limit definition you've been given very carefully. Then think about it very carefully. Then do lots of examples and think very carefully about what is happening in them.

4. so E is just an arbitrary value then, where in this case it's greater than 0, but what is N(E)?
from the below quote i can gather that 1/E will always be a positive value, but why is N(E) > 1/E?

Quote:
E>0 given
take N(E) s.t. N(E) > 1/E

Why do you think you can do that? If you can tell me this I will continue with my explanation, if you need it

5. Originally Posted by mathmonster
so E is just an arbitrary value then, where in this case it's greater than 0, but what is N(E)?
from the below quote i can gather that 1/E will always be a positive value, but why is N(E) > 1/E?
NO! By definition E is always greater than 0. That is because it is in some sense the distance between the limit and the numbers of the sequence. distances cannot be negative.

E>0 given
take N(E) s.t. N(E) > 1/E

Why do you think you can do that? If you can tell me this I will continue with my explanation, if you need it
That is the Archimedian property for real numbers.

It allows me to choose a natural number N(E) for a given E, such that such that it is greater than $\displaystyle \frac1{E}$

6. Originally Posted by Isomorphism
NO! By definition E is always greater than 0. That is because it is in some sense the distance between the limit and the numbers of the sequence. distances cannot be negative.

That is the Archimedian property for real numbers.

It allows me to choose a natural number N(E) for a given E, such that such that it is greater than $\displaystyle \frac1{E}$
ok so i get the first bit, i understand that now.
but, why are u choosing the natural number N(E)? could it possibly be called something different, like r, or does it have to be called N(E) and if so, why?

7. Originally Posted by mathmonster
ok so i get the first bit, i understand that now.
but, why are u choosing the natural number N(E)? could it possibly be called something different, like r, or does it have to be called N(E) and if so, why?
Yes you can call it whatever you want, however N(E) is a good choice since it reminds you that your choice depends on E. N(E) means the natural number is a function of E. Actually many books on analysis do not use that notation. They simply refer to it as N and it works fine too.

I hope you know what I mean when I say "Archimedian property". Pay attention to this small things that are used without explicit mentioning.
And follow Mr.F's words, and do a lot of examples and understand how you are approximating a given sequence with a limit.

As for the notation confusion, I hope it is clear. N(E) is just a natural number for a given E.

8. Originally Posted by mathmonster
so E is just an arbitrary value then, where in this case it's greater than 0, but what is N(E)?
from the below quote i can gather that 1/E will always be a positive value, but why is N(E) > 1/E?

Quote:
E>0 given
take N(E) s.t. N(E) > 1/E

Why do you think you can do that? If you can tell me this I will continue with my explanation, if you need it
N = N(E) emerges from the particular problem being considered - but you sometimes have to be quite clever to get it .....

N(E) = 1/E is N as a function of E emerged for this particular problem. If you take a value of n greater than or equal to N(E), your difference |1/n - 0| will be less than or equal to E.

Check how all this works with some concrete values of E ...... Execute the advice I gave at the end of my first reply.