Please, show that the sum and the product of two Cauchy sequences are Cauchy sequences.

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l3n4bananau6j · Accepted Answer

Let an and bn be cauchy sequences. We have to show that an+bn and anbn are also cauchy sequences.1) Since an and an are cauchy sequences, therefore, for ξ&amp;gt;0 there exist positive integers m1 and m2 such that|an−am1|&amp;lt;ξ2 ∀n≥m1|bn−bm2|&amp;lt;ξ2 ∀n≥m2If m=max[m1,m2] then |an−am|&amp;lt;ξ2 and |bn−bm|&amp;lt;ξ2 ∀n&amp;gt;mNow, |(an+bn)−(am+bm)|=|(an−am)+(bn−bm)|≤|an−am|+|bn−bm|&amp;lt;ξ2+ξ2=ξ ∀n&amp;gt;m|(an+bn)−(am+bm)|&amp;lt;ξ ∀n≥mThus, an+bn is a cauchy sequence2) Since every cauchy sequence is bounded, there exist a positive number k such that |an|≤k ∀nSince we proved that (1) are cauchy sequences, we can find positive integer m, which is suitable|bn−bm|&amp;lt;ξ2k ∀n≥m|an−am|&amp;lt;ξ2(|bm|+1) ∀n≥mNow, |anbn−ambm|=|an(bn−bm)+bm(an−am)|≤|an||an−bm|+|bm||an−am|&amp;lt;ξ2+ξ2(|bm||bm|+1) ∀n≥m|anbn−ambm|&amp;lt;ξ ∀n≥m|bm||bm|+1&amp;lt;1Hence, it&#039;s a cauchy sequence.

Please, show that the sum and the product of two

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