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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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 Fall '10
 Prof.KatrinWehrheim
 Sets, Upper Bound Property

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