Chapter 7: Q. 01 (page 603)
True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: If the tail of a sequence converges, the sequence converges.
(b) True or False: If and are two divergent sequences, then the sequence diverges.
(c) True or False: If is a convergent sequence of rational numbers, it must converge to a rational number.
(d) True or False: Every convergent sequence is bounded.
(e) True or False: Every bounded sequence is convergent.
(f) True or False:
(g) True or False: Every increasing sequence of negative numbers converges.
(h) True or False: Every increasing sequence of positive numbers converges.
Short Answer
(a) True.
(b) False.
(c) False.
(d) True.
(e) False.
(f) True.
(g) True.
(h) False.