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4130hw2 - < x 2< 2 is bounded above but has no least...

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4130 HOMEWORK 2 Due Thursday February 11 (1) In this exercise, we will show that the ordered field Q is not complete. (a) Suppose q Q and q 2 < 2. Let n N and suppose that n > max { 2 | q | +1 2 - q 2 , 1 } . Show that ( q + 1 n ) 2 < 2. (b) Suppose r Q and r 2 > 2. Let n N and suppose n > 2 r r 2 - 2 . Show that ( r - 1 n ) 2 > 2. (c) Using the results of (a) and (b), together with the fact that there is no s Q with s 2 = 2 (do not prove this), show that Q is not complete. (Hint: show that S = { x Q : 0
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Unformatted text preview: < x 2 < 2 } is bounded above but has no least upper bound.) (2) By looking in some books or on the internet, find two examples of ordered fields other than Q and R , including one which does not satisfy the archimedean property. (3) Show that any convergent sequence of rational numbers has a unique limit. (4) Section 2.1.3 # 1. (5) Section 2.1.3 # 8. 1...
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