Any help on this? I studied some more today but it's still a little fuzzy :/
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?
Its most easily understood with an example
take the sequence
Then the sup of is the least upper bound, which would be 7. Basically you look at the entire 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 is a tail
is a tail
is a tail
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 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
sup of is 5
but lim sup of is 1
These same analogies can be extended for lim inf and inf.