1. Find an example of each of the following. (a) A convergent sequence of rational numbers having an irrational limit. (b) A convergent sequence of irrational numbers having a rational limit. 2. Given...

1. Find an example of each of the following.

(a) A convergent sequence of rational numbers having an irrational limit.

(b) A convergent sequence of irrational numbers having a rational limit.

2. Given a sequence (s_n) and given k ∈ N, let (t_n) be the sequence defined by t_n
= s_n
+ k. That is, the terms in (t_n) are the same as the terms in (s_n) after the first k terms have been skipped. Prove that (t_n) converges iff (s_n) converges, and if they converge, show that lim t_n
= lim s_n. Thus the convergence of a sequence is not affected by omitting (or changing) a finite number of terms.