Show directly from the definition that if&nbsp;(xn)&nbsp;and&nbsp;(yn)&nbsp;are Cauchy sequences, then&nbsp;(xn+yn)&nbsp;and&nbsp;(xnyn)&nbsp;are Cauchy sequences.&nbsp;If&nbsp;xn&nbsp;=n, show that&nbsp;(xn)&nbsp;satisfies&nbsp;lim|xn+1–xn|=0, but that it is not a Cauchy sequence.&nbsp;Let p be a given natural number and give an example of a sequence&nbsp;(xn)&nbsp;that is not a Cauchy sequence, but that satisfies&nbsp;lim|xn+p–xn|=0.

Question

Show directly from the definition that if&amp;nbsp;(xn)&amp;nbsp;and&amp;nbsp;(yn)&amp;nbsp;are Cauchy sequences, then&amp;nbsp;(xn+yn)&amp;nbsp;and&amp;nbsp;(xnyn)&amp;nbsp;are Cauchy sequences.&amp;nbsp;If&amp;nbsp;xn&amp;nbsp;=n, show that&amp;nbsp;(xn)&amp;nbsp;satisfies&amp;nbsp;lim|xn+1–xn|=0, but that it is not a Cauchy sequence.&amp;nbsp;Let p be a given natural number and give an example of a sequence&amp;nbsp;(xn)&amp;nbsp;that is not a Cauchy sequence, but that satisfies&amp;nbsp;lim|xn+p–xn|=0.

pypberissootcu · Accepted Answer

1) We have to show that xn+yn and xnyn are Cauchy sequences if xn and yn are Cauchy sequenceWe know that these sequences are said to be Cauchy sequences if for given t&amp;gt;0, there exists a positive integer &quot;m&quot;, such that |an−am|&amp;lt;t∀n≥mFirst term let you prove that (xn+yn) is a Cauchy sequence.Given that xn and yn are CAuchy sequences. Let t&amp;gt;0 be any given numberPositive integers: m1 and m2|xn−xm|&amp;lt;t2∀n≥m1and |yn−ym|&amp;lt;t2∀n≥m2Let m=max[m1,m2]|xn−xm|&amp;lt;t2 and |yn−ym|&amp;lt;t2∀n≥m∴|(xn+yn)−(xm+ym)|=|(xn−xm)+(yn−ym)|≤|xn−xm|+|yn−ym|&amp;lt;t2+t2=t∴|(xn+yn)−(xm+ym)|&amp;lt;t∀n≥m∴(xn+yn) is a Cauchy sequence Now, prove that (xnyn) is a Cauchy sequence|xnyn−xmym|=|xnyn−xnym+xnym−xmym||xn(yn−ym)+ym(xn−xm)|≤|xn||yn−ym|+|xn−xm||ym| (1)Since xn is a Cauchy sequence and every Cauchy sequence is bounded∴xn is boundedTherefore, a number k&amp;gt;0 such that |xn|&amp;lt;k∀n∈NAlso yn is a Cauchy sequence Therefore, positive integer m, such that |yn−ym|&amp;lt;t2k∀n≥m1 (2)Now, same for xn Cauchy sequence Positive integer m2 such that |xn−xm|&amp;lt;t2|ym|∀n≥m2Now, let m=max[m1,m2]|xn−x|&amp;lt;t2|yn| and |yn−ym|&amp;lt;t2k∀n≥m (3)Using equation (3) and |xn|&amp;lt;k in equation (1)|xnyn−xmym|&amp;lt;k×t2k+t2|ym|×|ym|∀n≥m⇒|xnyn−xmym|&amp;lt;t∀n≥m∴(xnyn) is a Cauchy sequence

Jayda Burch · Accepted Answer

Show directly from the definition that if

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Answer & Explanation

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