Limit of sequence proof (elementary analysis)

[ironman1478]

Homework Statement

Consider three sequences a(n), b(n) and s(n) such that an a(n) <= s(n) <= b(n) for all n and lim(a(n)) = lim(b(n)) = s. Prove that lim(s(n)) = s

(n) is a subscript

Homework Equations

the book i am doing this problem from is elementary analysis, the theory of calculus and it defines the definition of a limit for a sequence as

|s(n) - s| < E
so its sort of like an epsilon delta proof

The Attempt at a Solution

i really don't know how to do this problem. i tried to do it, but was totally wrong so i was wondering if you guys could help me understand the solution. i won't post the whole thing, just the solution up to the part that i didnt understand.


book proof
let E > 0. to prove this, we need to show s-E < s(n) < s + E for large n. since lim (a(n)) = s, there exists N(1) such that |a(n) - s| < E for n > N(1). in particular
n > N(1) implies s - E < a(n)

this is the part that i don't get. i can't figure out |a(n) - s| < E and n > N(1) implies that. the only way that makes sense is if we said that (a(n) - s) < 0 therefore |a(n) - s| = - (a(n)-s) and then we can get what n > N(1) implies. however, if this is true, why can we assume a(n) - s < 0 or am i just misunderstanding what the thing is saying

 

[Fredrik]

So the detail you're currently stuck on is to prove that if $|a_n-s|<\varepsilon$ for all $n\geq N_1$, then $s-\varepsilon<a_n$ for all $n\geq N_1$?

All you have to do is to note that for all real numbers x, we have x≤|x|. In particular, we have $s-a_n\leq|s-a_n|=|a_n-s|<\varepsilon$.

By the way, if you don't want to use LaTeX, you can still type some symbols like ε,≤,→. You can find them to the right of the input field when you're typing a post. You can also use vBulletin's sub tags to get an index, like this: a_n. (Hit the quote button to see what I typed).

 

[ironman1478]

ah thank you. i didnt think of that fact. i will try to use LaTex more.

also, i just have a general question. is it normal to be awful at these kind of proofs? i was able to get all of the other proofs in the book not related to epsilon delta stuff, but for some reason i just can't figure these kinds out and i always get stuck on them (happened in my calc class and its happening when i go through this book).

 

[Fredrik]

Yes, it's normal to be awful at them, for a while at least. I think most people don't get past the "awful" stage until some time during their first analysis course, and I think most people who are good at them have taken a topology course as well.

 

[ironman1478]

Yes, it's normal to be awful at them, for a while at least. I think most people don't get past the "awful" stage until some time during their first analysis course, and I think most people who are good at them have taken a topology course as well.

lol that makes me feel better. i am trying to go through this book before taking analysis 1 so i don't get destroyed, but its a real pain since i don't have any proof experience.