3 4 Find an example of a sequence of real numbers satisfying each set of

3 4 find an example of a sequence of real numbers

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4.3-4* Find an example of a sequence of real numbers satisfying each set of properties. 20
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1. Cauchy, but not monotone. 2. Monotone, but not Cauchy. 3. Bounded, but not Cauchy. Solutions: 1. ( ( 10 n n ) is Cauchy, but not monotone. 2. (1 , 2 , 3 , 4 , ... ) is monotone, but not Cauchy. 3. (1 , 0 , 1 , 0 , 1 , 0 , ... ) is bounded, but not Cauchy. 4.3-6* Let ( a n ) and ( b n ) be monotone sequences. Prove or give a counterex- ample, 1. The sequence ( c n ) given by c n = ka n is monotone for any k R . 2. The sequence ( c n ) given by c n = a n /b n is monotone, where b n = 0 for all n N . Solutions: 1. Yes. ( c n ) = ( ka n ) is monotone. In fact, if k > 0, we have that ka n ka n +1 if and only if a n a n +1 ; and that ka n ka n +1 if and only if a n a n +1 . Therefore, ( ka n ) is monotone if and only if ( a n ) is monotone. Similarly we can consider the case k < 0 and k = 0. 2. No. Here is a counterexample: ( a n ) = (1 , 2 , 2 , 3 , 3 , 4 , 4 , .... ) and ( b n ) = (1 , 1 , 2 , 2, 3 , 3 , 4 , 4 , ... ) are monotone. But the sequence a n b n = 1 , 2 , 1 , 3 2 , 1 , 4 3 , ... is not monotone. 4.3-9* Suppose x > 0 . De fi ne a seqeunce ( s n ) by s 1 = k and s n +1 = s 2 n + x 2 s n for n N . Prove that for any k > 0 , lim s n = x . Solutions: ( s n ) is bounded below We claim that s n x for all n 2, i.e., s 2 n x 0 for all n 2. In fact, 21
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s 2 k +1 x = s 2 k + x 2 s k 2 x = s 4 k + 2 s 2 k x + x 2 4 s 2 k x = s 4 k + 2 s 2 k x + x 2 4 s 2 k x 4 k 2 = ( s 2 k 2) 2 4 k 2 0 . holds for any k 1. ( s n ) is decreasing : We want to prove that ( s n ) is decreasing for n 2: s n s n +1 0 for all n 2. In fact, s n s n +1 = s n x 2 n + x 2 s n = s 2 n x 2 s n 0 . Here for the last inequality, we have used the ineqyality s 2 n x 0 proved above.
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  • Natural number, lim sn

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