For any real sequence define a) Prove that if lim infkxk > x for some x R, then xk > x for k large. b)

For any real sequence define

a) Prove that if lim infk†’ˆžxk > x for some x ˆˆ R, then xk > x for k large.
b) Prove that if xk †’ x as k †’ ˆž, for some x ˆˆ R, then lim infk†’ˆž xk = x.
c) If ak > 0 for all k ˆˆ N, prove that

d) Prove that if bn ˆˆ R {0} and |bn+1/bn| †’ r as n †’ ˆž, for some r > o then |bn]1/n †’ r as n †’ ˆž.

lim infAk := lim infxk lim inf ak+1 < lim inf < lim sup t/ak lim sup akti