1) Let such that then for all there exists an such that if then let and so for all , and so . In a similar fashion you prove that can't be negative.
2) Let
(a) Suppose that {an} is a convergent sequence of points all in [0, 1] (ie.
an∈[0, 1] for all n). Prove that limit of an as n-->∞ is also in [0, 1].
(b) Find a convergent sequence {an} of points all in (0, 1) such that limit of an as n-->∞
is not in (0, 1).