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 .