1. Prove that Q is dense in R with the usual absolute value metric. [If a metric space (X, d ) has a countable subset that is dense, then X is said to be separable. Thus in this exercise you are to...

1. Prove that Q is dense in R with the usual absolute value metric. [If a metric space (X, d ) has a countable subset that is dense, then X is said to be separable. Thus in this exercise you are to show that R is separable.]

2. Prove that {(x, y): x ≠ 0 and y ≠ 0} is dense in

²
with the usual Euclidean metric.

3. Mark each statement True or False. Justify each answer.

(a) If (s_n) is a sequence and s_i
= s_j, then i = j.

(b) If s_n
→ s, then for every ε > 0 there exists N ∈
 such that n ≥ N implies | s_n
– s |

(c) If s_n
→ k and t_n
→ k, then sn = tn for all n ∈
.

(d) Every convergent sequence is bounded