Real analysis: Limit superior proof

[kingwinner]

Homework Statement

Definition:
Let (a_n) be a sequence of real numbers. Then we define
lim [sup{a_n: n≥k}] = lim sup a_n
k->∞

(note: sup{a_n: n≥k} = sup{a_k,a_k+1,a_k+2,...} = b_k
(b_k) is itself a sequence of real numbers, indexed by k)

Theorem:
Let a=lim sup a_n.
Then for all ε>0, there exists N such that if n≥N, then a_n<a+ε

Homework Equations

N/A

The Attempt at a Solution

I am trying to prove the theorem.
Proof:
Let b_k=sup{a_n: n≥k}.
By the definition of "a" as a limit of supremums (b_k->a), we have that
for all ε>0, there exists an integer N such that if k≥N, then
|sup{a_n: n≥k} - a| = |b_k -a|< ε
=> sup{a_n: n≥k} < a + ε
By definition of an upper bound, sup{a_n: n≥k} ≥ a_n if n≥k.
So the above shows that if k≥N and n≥k, then a_n< a+ ε.

Now I am stuck...how should I continue? (I need to prove that: for all ε>0, there exists N such that if n≥N, then a_n<a+ε. But from my work so far, I'm really feeling hopeless)

There are so many different subscripts (e.g. n and k) to keep track of that I am really puzzled and frustrated now.

Can someone please help me out and show me the correct way to prove the theorem?
Any help is very much appreciated!

 

[mighty2000]

Thank you for your post. I can see that you have made a good start in your proof of the theorem. Let me try to guide you towards the next steps.

First, let's recall what we are trying to prove: for any given ε>0, we want to show that there exists an integer N such that if n≥N, then an<a+ε. In other words, we want to show that eventually, all terms in the sequence an will be less than a+ε.

Now, you have already shown that for any given ε>0, there exists an integer N such that if k≥N, then sup{an: n≥k} < a+ε. But this is not quite what we want, because the supremum of the terms in the sequence an is not necessarily the same as the terms themselves.

So, how can we use the fact that the supremum is less than a+ε to show that the terms in the sequence an are also less than a+ε? Think about what the supremum represents - it is the smallest upper bound for the terms in the sequence an. Can you see how this can help you prove the theorem?

I hope this helps you make progress in your proof. If you are still stuck, feel free to reply to this post and I will try to guide you further. Good luck!