Characterisation of the limit superior and limit inferior

Question

in my notes (University 1st year Analysis) is the following proposition :

proposition !

with the proof

proof !

I don't understand what it means for the set to be finite/infinite and I am therefore a little hazy with the steps in the proof. Any simplification/justification would be much appreciated.

a set is finite if it has finitely many elements. A set is infinite if it has infinitely many elements. Are you asking for that? — user126154, Mar 04 '14 at 11:45
@user127700 : Is there a particular step which troubles you? — Patrick Da Silva, Mar 04 '14 at 11:49
@PatrickDaSilva I struggle particularly with the first paragraph of (<=) step, ie that it follows the set is infinite. — user127700, Mar 04 '14 at 11:51

score 2 · Answer 1 · answered Mar 04 '14 at 11:58

In plain english, this means that $\alpha$ is the limes superior of the sequence $a_n$ exactly if for every $\epsilon > 0$

infinitely many of the $a_n$ are larger than $\alpha - \epsilon$.
but only finitely many of the $a_n$ are larger than $\alpha + \epsilon$

In other words, for every non-empty interval $(\alpha-\epsilon,\alpha+\epsilon)$ around $\alpha$

infinitely many of the $a_n$ lie within the interval.
but only finitely many of the $a_n$ lie to the right of the interval

(1) means that $a_n$ is a limit point of the sequence, and (2) means that it's the largest limit point.

Characterisation of the limit superior and limit inferior

1 Answers1