From various textbooks, I have read that:

(1) When a sequence sn as n one writes that the xlimsn= if for M>0 , N such that if n > N then sn > M.

(2) When a sequence sn as n one writes that the nlimsn= if for M0 , N such that if n > N then sn

I think I can use these two definitions to help me prove the following theorem given that the proof requires me to consider a case where the limit mirrors (1) and the other case mirrors (2). However, these definitions might be unnecessary. what are the proofs for the two cases in the following theorem.

Let &'n', { X' be sequences of real numbers where lim y. A ? No for some No. If lim Xu - Xin.` nico In = to E Yak Yay, for n > bo Un - 4_ _ = 100 or - 10 1i'm Xn _ lim Xn - X` - !) Yn "in too Yo - Yo - 1