From various textbooks, I have read that:

(1) When a sequencesnasnxone writes that thexlimsn=if forM>0,Nsuch that if n > N thensn> M.

(2) When a sequencesnasnxone writes that thenlimsn=xlimsn=if forM0,Nsuch that if n > N thensn

I think I canusethese two definitionstohelpme prove thefollowingtheorem given that the proof requires metoconsideracasewherethelimitmirrors (1)andthe othercasemirrors (2). However, these definitions might be unnecessary. whatarethe proofsforthe two casesinthefollowingtheorem.

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