What is the relationship between lim sup and lim inf in a convergent sequence?

[happyg1]

Hi,
I'm having trouble understanding why a sequence converges if and only if lim sup=lim inf. I think about the sequence {1/n} this sequence converges to 0, but the lim sup is 1. How is the limsup 0? What am I missing?
HELP
CC

 

[TD]

Could you please be a bit more clear? What do you mean exactly with "lim sup" and "lim inf". The 'limit' of the 'supremum' and 'infinity'?

As you say, {1/n} converges to 0 (i.e. the sequence goes to 0 as n goes to infinity), but what do you mean with "limsup"?

 

[happyg1]

that's exactly my quandry.
There's a theorem in the book that says
"If {s} is a sequence of real numbers, and if limsup s = liminf s =L where L is in R, then {s} is convergent and lim s = L."

So I don't understand how in the case of {1/n} this holds. 1/n is a convergent sequence and it's limit is 0, but looking at the theorem there, it seems that limsup s must be zero also. But limsup {1/n} is 1. What am I missing?

HELP

 

[cmurphy]

lim sup is the limit of the supremum for large N, right? lim sup sn = lim as N-> infinity of the set {sn : n > N}.

So begin listing some terms of sn: 1, 1/2, 1/3, 1/4, 1/5, 1/6, ...

If N = 1, then n must be at least 2. So we ignore the first term, and look at the sequence 1/2, 1/3, 1/4, 1/5, 1/6, ... The sup in this case is 1/2.

If N = 2, then n must be at least 3. So we ignore the first two terms, and look at the sequence 1/3, 1/4, 1/5, 1/6, 1/7, ... The sup in this case is 1/3.

If N = 3, then n must be at least 4. So we ignore the first three terms, and look at the sequence 1/4, 1/5, 1/6, 1/7, 1/8. Now the sup is 1/4.

We must look at the lim sup. So we need the LIMIT of the supremum. Of course the limit of the supremum is going to be 0.

Similarly, the lim inf is 0:

If N = 1, then n must be at least 2, so we look at the sequence 1/2, 1/3, 1/4, 1/5, ...
and you can see that the lim inf is 0.

Thus lim inf sn = lim sup sn = 0.

I think that's the right way to look at it!

 

[happyg1]

OOOOOOOOOOOO
Now I see what they mean. The fog has lifted!
THANKYOU VERY VERY MUCH

 

[HallsofIvy]

lim sup is the limit of the supremum for large N, right?

More correctly, "lim sup" of a sequence is the supremum (upper bound) of all subsequential limits. And a sequence converges if and only if all subsequences converge to the same limit.

I think about the sequence {1/n} this sequence converges to 0, but the lim sup is 1.

You are confusing "lim sup" of a sequence with "sup" of a set. The supremum of the set {1/n} is the largest member, 1, but since the sequence converges to 0, all subsequences also converge to 0. The set of all "subsequential limits" is {0} and both sup and inf of that set is 0.

 

[NascentOxygen]

Posters have been patiently waiting over 9 years for that assistance, HallsofIvy.

I hope it's appreciated; if not by them, then at least by others now that you have revitalised the thread.

 

[HallsofIvy]

How in the world did I manage to get into "2005"? Time travel?