A

14. Let {a_n} be a sequence of real numbers. Which of the following statements must be true?

I. If {a_n} is unbounded, then every subsequence of {a_n} diverges.

II. If {a_n} is diverges, then every subsequence of {a_n} also diverges.

III.If {a_n} is converges, then every subsequence of {a_n} also converges.

A. I and II only	B. I and III only	C. II and III only
D. I, II, and III	E. The correct answer is not given by A, B, C, or D.

Solution: The answer is E

Statement I: Not true.

Reason:

The unbounded sequence, {a_n} = 0,1,0,2,0,4,0,6,0,… , has {b_n} = 0,0,0,0,… as a convergent subsequence.

Statement II: Not true.

Reason:

The divergent sequence, {a_n} = 1,-1,1,-1,1,-1,… , has {b_n} = 1,1,1,1,1,… as a convergent subsequence.

Statement III: True. This is a standard result on sequences.

The only statement that is true is statement III, therefore the correct answer is not given by A, B, C, or D.