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Unformatted text preview: Q in R . (b) Use denitions and ideas from your lecture notes to prove that I is dense in R . (c) If D is a dense subset of R , prove that D must be innite. Proof by contradiction is useful here. 5. Pages 106-111 # 1(iv), (vi) (Just do the calculations; no proofs); 3(i), (ii); 10(d); 13; 17(a); 18; 19. 6. Give a proof of Theorem 1 on pages 98-99 that does not involve contradiction. 7. Let f ( x ) = x + 3 2 x . Prove that lim x a f ( x ) = f ( a ) for all a 6 = 0. 8. Prove carefully that for all n N , lim x n [ x ] does not exist. 9. Let D R and let f : D R be a function. Given a real number a , if lim x a f ( x ) = l and l > 0, prove there exists a > 0 such that f ( x ) > 0 for all x D ( a-,a + ) but x 6 = a . 10. Page 117 # 1(i); 3; 4....
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- Winter '08