homework_3 - 100000000 . On the other hand, the terms of...

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Homework 3 5.6. If S and T are bounded then the result holds by Exercise 4.7(a) (proved in class). If S is not bounded above, then T is not bounded above because S T . Hence, if sup S = then sup T = and so sup S sup T in all cases. Similarly, if S is not bounded below, then T is not bounded below. Hence, if inf S = -∞ then inf T = -∞ and so inf T inf S in all cases. Finally, we have inf S sup S in all cases because we order R ∪ {-∞ , ∞} by -∞ < x < for all x R . 7.4. (a) Consider the sequence x n = 1 n 2. If any number in this sequence were rational, then we could write 1 n 2 = p q 2 = np q where np and q are integers. Hence, 2 would be rational which is a contradiction. This is a sequence of irrational numbers. However, the terms of the sequence become smaller and smaller, so the limit of the sequence is 0, which is a rational number. (b) Consider the sequence r n consisting of decimal approximations to n digits of 2. Any finite decimal approximation is rational; for example r 8 = 1 . 41412136 = 141412136
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Unformatted text preview: 100000000 . On the other hand, the terms of this sequence tend to 2 by denition, which is not rational. 8.2. (a) lim n n n 2 +1 = 0. Let &amp;gt; 0 and choose N = 1 . Then, n &amp;gt; N implies n &amp;gt; 1 , and 1 n &amp;gt; 0 for all n N , so n + 1 n &amp;gt; n &amp;gt; 1 . This is n 2 + 1 n &amp;gt; 1 so | n n 2 + 1 | = n n 2 + 1 &amp;lt; . (c) lim n 4 n +3 7 n-5 = 4 7 . Let &amp;gt; 0 and choose N = 1 49 ( 41 + 35). Then, n &amp;gt; N implies n &amp;gt; 1 49 ( 41 + 35) so (49 n-35) &amp;gt; 41 and &amp;gt; 41 (49 n-35) = (28 n + 21)-(28 n-20) 7(7 n-5) = 7(4 n + 3)-4(7 n-5) 7(7 n-5) 1 2 so | 4 n + 3 7 n-5-4 7 | = 7(4 n + 3)-4(7 n-5) 7(7 n-5) &amp;lt; . (e) lim n 1 n sin n = 0. Let &amp;gt; 0 and choose N = 1 . Then, n &amp;gt; N implies n &amp;gt; 1 and so we have 1 n &amp;lt; . Hence, | 1 n sin n | = 1 n | sin n | 1 n &amp;lt; because | sin n | 1 for all n N ....
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homework_3 - 100000000 . On the other hand, the terms of...

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