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MIT18_100BF10_pset1

# MIT18_100BF10_pset1 - property 4(a Prove that the ﬁeld Q...

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18.100B : Fall 2010 : Section R2 Homework 1 1 n Due Tuesday, September 14, 11am Reading: Thu Sept.9 : ordered sets and ﬁelds, Rudin 1.1-31 . 1 . Suppose that S is a set and p is a relation on S with the following properties: For all x S , x p x . For all x, y S , if x p y and y p x then x = y . For all x, y, z S , if x p y and y p z , then x p z . Deﬁne a new relation on S by x y iff x p y and x = n y . Does this deﬁne an order conforming to Deﬁnition 1.5 in Rudin? If so, prove it; if not, exhibit a counterexample. 2 . Exercise 6, p. 22 of Rudin. [Here b > 1 is an element of R and you may use the ‘deﬁnition’ of R as ordered ﬁeld with least upper bound property. Then recall that y = x is deﬁned n as solution of y = x, y 0 .] 3 . (Exercise 9 p. 22 of Rudin – lexicographic order) For complex numbers z = a + bi C and w = c + di C deﬁne “ z < w ” if either a < c or if ( a = c and b < d ). Prove that this turns C into an ordered set. Is this an ordered ﬁeld? Does it have the least-upper-bound

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Unformatted text preview: property? 4 . (a) Prove that the ﬁeld Q of rational numbers has the Archimedean property. (b) Compare the least upper bound property with the Archimedean property – which one is ‘stronger’? Why? 5 . Review the logic of a proof by induction. (The – not always reliable – wikipedia gives a good explanation in this case.) (a) Prove that (1 + 2 + ··· + n ) 2 = 1 3 + 2 3 + ··· + n 3 for each n ∈ N . (b) Find a proof of the Bernoulli inequality: (1 + x ) n ≥ 1 + nx for all x ∈ R , x ≥ − 1 and n ∈ N , n ≥ 2 . (not for credit) Show that strict inequality (1 + x ) n > 1 + nx holds unless x = 0 . MIT OpenCourseWare http://ocw.mit.edu 18.100B Analysis I Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT18_100BF10_pset1 - property 4(a Prove that the ﬁeld Q...

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