LimSup of a Sequence: Existence & Unbounded Cases

[C.E]

1. Let a_n be a sequence in R. Define $$\limsup_{n \to \infty} a_n$$. Explain why it exists including the cases when the sequence is unbounded.

3. I did the following, is it ok?

definition:

$$\limsup_{n \to \infty} a_n$$=$$\lim_{n \to \infty} sup_{k \geq n}a_k$$.

Is this right, I seem to have lost this part of my notes!

I know that when it is not bounded (above) the limsup is infinite. What I am really having

trouble with is explaining why it exists. What do you think they want me to say? (By the

way this question was worth 10 marks so they want a lot of detail).

 

[Billy Bob]

Let $$v_n=sup_{k \geq n}a_k$$

What kind of sequence is v_n? increasing, decreasing, bounded,...

 

[C.E]

Is v_n always a decreasing sequence? It may be bounded or it may not be. If it is bounded then by the completeness axiom its limit must exist, but what if it is not bounded?

 

[jbunniii]

Is v_n always a decreasing sequence? It may be bounded or it may not be. If it is bounded then by the completeness axiom its limit must exist, but what if it is not bounded?

If it's monotonically decreasing but unbounded, then for every positive B, there is an N such that

$$v_n \leq -B$$ for all $$n \geq N$$

in which case by definition we write

$$\lim_{n \rightarrow \infty} v_n = -\infty$$

 

[C.E]

Is it increasing then? If so why?