No theorem for limit inferior in textbook

Course: Intro to Real Analysis

I'm trying to prove , where and are bounded sequences. There are only theorems for limit superior in our chapter, so I was wondering if I could somehow manipulate the following theorem:

Let be a bounded sequence. Then the following properties hold:

(a) For every there exists a natural number such that implies that .

(b) For every and for every there exists an integer such that .

Can I say the following?

For every there exists a natural number such that implies that .

For every and for every there exists an integer such that .

1 Attachment(s)

Re: No theorem for limit inferior in textbook

Hi,

Yes. The attachment shows a proof of your statement; this is an exact analog for the corresponding statement about lim sup's. I think the "trick" about using limits when working with lim sup or lim inf is worth remembering.

Attachment 30466

Re: No theorem for limit inferior in textbook