HW 26 extra problems

Prove that a nonzero geometric series \sum (a₀ r^k) converges iff |r|<1.

Prove Theorem 6.2.2 (The Comparison Test) without looking at your notes or the book's proof.

Let f : N --> Q be a bijection. Define a sequence {a_k} by a_k = f(k).

1. (a) Prove that every rational number is a limit point of {a_k}.

2. (b) Prove that every real number is a limit point of {a_k}.

Let A, B, C be subsets of R. Show that if B is dense in C and every point in B is a limit point of A, then every point in C is a limit point of A.