(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

N/A

3. 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 alimitof 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!

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# Real analysis: Limit superior proof

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