# Limsup and bounded above

1. Apr 5, 2012

### kingwinner

Claim (?):
limsup xk < ∞
k->∞

IF AND ONLY IF

the sequence {xk} is bounded above.

Does anyone know if this is true or not? (note that the claim is "if and only if")
If it is true, why?

Thanks!

2. Apr 6, 2012

### kingwinner

I think I'm OK with the direction:
limsup x_k < ∞ implies the sequence {x_k} is bounded above.
k->∞
(becuase if the sequence is not bounded above, then limsup=∞)

But is the converse direction also true?

The sequence {x_k} is bounded above

implies

limsup x_k < ∞ ???
k->∞

3. Apr 6, 2012

### chiro

Hey kingwinner.

Think about the idea of the lowest upper bound. Suppose there is a value that is above this upper bound. Then this means the lowest upper bound is that new value.

Now imagine that all values are finite. What does this say about the lowest upper bound? What happens if one value is infinite (positive infinity)?

4. Apr 6, 2012

### HallsofIvy

Staff Emeritus
"lim sup xn" is, by definition, the supremum of the set of all subequential limits. If it were not finite, then, given any M, there would have to exist subsequences of {xn having limit larger than M so M could not an upper bound of {xn}.

5. Apr 6, 2012

### kingwinner

Does this imply that
the sequence {x_k} is bounded above => limsup x_k < ∞ ?

Are you using contrapositive?

Last edited: Apr 6, 2012
6. Apr 6, 2012

### HallsofIvy

Staff Emeritus
Yes, that is proving "if a then b" by proving the contrapositive, "if b is not true then a is not true".

7. Apr 6, 2012

### kingwinner

Is it possible to prove "directly" that

sequence {x_k} is bounded above => limsup x_k < ∞ ??

8. Apr 12, 2012

### Congruent

Yes. Using the definition above, let $\{x_{k_n}\}_{n=1}^{\infty}$ be any convergent subsequence. By assumption, there is an $M > 0$ so that $x_k \le M$. In particular, $M \ge x_{k_n}$. Thus $M \ge \lim_{n \to \infty} x_{k_n}$. This shows that M is an upper bound to any subsequential limit. By the definition HoI gave for the limsup as the supremum of all subsequential limits, this shows that limsup x_k < \infty.

9. Apr 14, 2012

### kingwinner

How about this? limsup means the limit of the supremum of terms with greater and greater index. If the whole set is bounded above, then sup_{k>0} is finite, right? sup(k>0), sup(k>1), sup(k>2),... forms a decreasing sequence right? The limit is either well defined real number or -infinity, isn't it?

So in some problem, say if we were to show that limsup x_k < infinity, it suffices to prove that sequence {x_k} is bounded above, and vice versa, is that correct?

Thanks.