Understanding Limit Supremum and Limit Infimum

It's easy for me to say that the Limit Supremum is the "largest subsequential limit" and the Limit Infimum is the "smallest subsequential" limit, but that's about as far as my understanding goes.

What's a good, intuitive way to understand these concepts?

Re: Understanding Limit Supremum and Limit Infimum

Any help on this? I studied some more today but it's still a little fuzzy :/

Re: Understanding Limit Supremum and Limit Infimum

Its most easily understood with an example

take the sequence $\displaystyle S_n = \{5,6,7,2,4,-1,1,-1,1,-1,1,...\}$

Then the sup of $\displaystyle S_n$ is the least upper bound, which would be 7. Basically you look at the entire $\displaystyle S_n$ and say 7 is the smallest number which is an upper bound for this set.

But for lim sup, you basically take a look at the tails of this sequence and see what the least upper bound is.

By tail i mean $\displaystyle S_n = \{5,6,7,2,4,-1,1,-1,1,-1,1,...\}$ is a tail

$\displaystyle S_n = \{6,7,2,4,-1,1,-1,1,-1,1,...\}$ is a tail

$\displaystyle S_n = \{7,2,4,-1,1,-1,1,-1,1,...\}$ is a tail

$\displaystyle S_n = \{2,4,-1,1,-1,1,-1,1,...\}$ is a tail

im essentially removing the first element of the sequence, the second element, and third, and fourth etc... and taking the sup of those sequences.

I do this an infinite amount of times.

So for our sequence $\displaystyle S_n$ the lim sup is 1.

The sup of the first tail is 7

sup of second tail is 7

third is 7

fourth is 4

fifth is 4

sixth is 1

seventh is 1

etc...

Basically if you have a finite number of "big" numbers in the sequence, then the sup of the sequence only gives a distorted "least upper bound".

but lim sup gives you the sup if you throw away the finite number of "big" numbers in the sequence

in the sequence $\displaystyle P_n = \{5,1,1,1,1,1,1,1,1,1,...\}$

sup of $\displaystyle P_n$ is 5

but lim sup of $\displaystyle P_n$ is 1

These same analogies can be extended for lim inf and inf.