Neighborhoods and Limit Points

Definition 1) Let x be any real number. A neighborhood of x with positive radius r is the set Definition 2) A deleted neighborhood of x, denoted N'(x;r), is the nbd N(x;r) with the point x itself removed Definition 3) Given a nonempty set S in R, a point x in R is said to be a limit point of S if, for each ε>0, the deleted nbd contains at least one point of S Bolzano-Weierstrass Theorem) If S is a bounded, infinite subset of R, then S has a limit point in R. pf) Since S is bounded, sup S and inf...