1) Rudin, Ch. 1, Problem 1: If r is rational (r not equal to 0) and x is irrational, prove that r+x and rx are irrational.
2) If (a_n), (b_n) are rational sequences, and lim a_n = a, lim b_n = b, show that: lim(a_n+b_n) = a+b, and lim(a_n b_n) = ab.
3) Show: If a_n converges to a, any subsequence (a_(n_k)) of (a_n) converges to a.
4) Let (b_n) be a Cauchy sequence of rational numbers such that b_n is not zero for any n. Show that if (b_n) does not converge to 0, then (b_n) has a multiplicative inverse in the set of Cauchy sequences.
5) Show that between any two distinct real numbers there are infinitely many rational numbers.