a c D

You are given three real numbers a, c, d, with c < a < d, and a set of real numbers, D,
such that the open interval (c, d) is a subset of D.
1. Suppose {an} is a sequence in D converging to a. Prove that there exists an index
N such that for all n ≥ N, {an} is in (c, d).
2. Prove that there is a subsequence of
{an} in (c, d) which converges to a.
3. You are given a real-valued function
f on D. Let g denote the restriction of
f to the open interval (c, d). Using the
sequential definition of continuity, show
that f is continuous at a iff g is
continuous at a.