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enter image description here Which among is the correct choice .and please explain it.

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2 Answers 2

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a_n =1/n makes{a_n} as Cauchy sequence f(x)=1/x^3 is uniformly continuous in (1,4) which is not a Cauchy sequence. Also , y_n = 1/ ( 1+( a_n)^2 = n^2/(n^2+ 1)(n^6) = n^8/(n^2+1)~n^6. Hence y_n is also not a Cauchy sequence . so neither {x_n} nor {y_n } be a Cauchy sequence in R.

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I assume "R" and "N" here mean $\mathbb{R}$ and $\mathbb{N}$?

In which a case a sequence is Cauchy iff it converges to some $p \in \mathbb{R}$ (without necessarily direct knowledge the value of $p$)

So just fix some $\{a_n\}$ convergent, $f(x)$ uniformly continuous for which $\{x_n\}, \{y_n\}$ do not converge.

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