Again using this theorem: Let E be a set of real numbers. Then p is an accumulation point of E iff the following condition holds:
there is a sequence (xn) of members of E, each different from p, such that (xn) converges to p.
How would you prove or gain the following characterization about closure?
(1) Let E be a set of real numbers. A real number p belongs to the closure of E iff there is a sequence (xn) of members of E converging to p.
(2) Conclude, in particular, that if E is a non-empty bounded subset of R and s = supE, then there is an increasing sequence (xn) of members of E converging to s.