1. Give an example of a sequence of positive rational numbers {n} 1 such that the...

 

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1. Give an example of a sequence of positive rational numbers {n} 1 such that the following statement holds: Vε > 0, ³n > 1 such that o̟n > ½ and 0 < | In - In+1| < ɛ. (2 points) 2. Give an example of a sequence of positive rational numbers {n} x₁ such that the following statement holds: VK > 0, n > 1 such that |In - In+1| > K. (2 points) 3. Give an example of a sequence of positive rational numbers {n} 1 such that the following statement holds: V positive rational number a there exists n ≥ 1 such that Ina and Qn+1 < a. (2 points) 4. Show that {n}x_₁ and {rn} ~ 2₁ are Cauchy-sequences of rational numbers, then {In+rn} is a Cauchy-sequence as well. n=1 (4 points) 1. Give an example of a sequence of positive rational numbers {n} 1 such that the following statement holds: Vε > 0, ³n > 1 such that o̟n > ½ and 0 < | In - In+1| < ɛ. (2 points) 2. Give an example of a sequence of positive rational numbers {n} x₁ such that the following statement holds: VK > 0, n > 1 such that |In - In+1| > K. (2 points) 3. Give an example of a sequence of positive rational numbers {n} 1 such that the following statement holds: V positive rational number a there exists n ≥ 1 such that Ina and Qn+1 < a. (2 points) 4. Show that {n}x_₁ and {rn} ~ 2₁ are Cauchy-sequences of rational numbers, then {In+rn} is a Cauchy-sequence as well. n=1 (4 points)

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1 Let an 0 choose n such that c Then for all n 1 we n nn1 e n n1 2 L... View the full answer