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(UGC CSIR-2015, DECEMEMBER, MATHEMATICAL SCIENCES)

  1. $f$ maps a bounded sequence to a bounded sequence.
  2. $f$ maps a Cauchy sequence to a Cauchy sequence.
  3. $f$ maps a convergent sequence to a convergent sequence.
  4. $f$ is uniformly continuous.

I choose all of the options as possible answers because the condition $\sup_{x\in \Bbb R}|f'(x)| \lt \infty$ forces $f$ to be uniformly continuous.(Because $f$ becomes Lipschitz and Lipschitz condition implies uniform continuity)

i.e. $\frac {|f(x)-f(y)|}{|x-y|} \le \sup_{x\in \Bbb R}|f'(x)|$ $ \forall x,y$.

Hence all other options are bound to be true.

Am I correct?

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  • 4
    $\begingroup$ correct. The condition even implies that $f$ is Lipschitz. $\endgroup$ – user295959 Dec 21 '15 at 8:41
  • $\begingroup$ $user295959 Exactly!! I forgot to put this above in the post. $\endgroup$ – Error 404 Dec 21 '15 at 8:42
  • $\begingroup$ I had asked this question on 21st December 2015 while @S.Panja-1729 had asked this question on 22nd December 2015. Then why is it so that my question is duplicate of his? $\endgroup$ – Error 404 Jan 3 '16 at 12:28
  • $\begingroup$ Which question was asked first is not the prime concern. If the newer question is better in itself or has received better answers, that's a reason to close the older as a duplicate of the newer. Here, I reversed the direction of the duplicate because the answer here is more comprehensive, the answer at the other question treats only the first point. $\endgroup$ – Daniel Fischer Jan 3 '16 at 12:49
  • $\begingroup$ Thank you @DanielFischer and I agree with you. $\endgroup$ – Error 404 Jan 3 '16 at 12:54
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All are correct.

  1. If $\{x_n\}_{n\in\mathbb N}\subset\mathbb R$ is bounded, i.e., $\lvert x_n\rvert\le M<\infty$, then $$ \lvert\,f(x_n)-f(x_1)\rvert=\lvert x_n-x_1\rvert\lvert f'(y_n)\rvert, $$ for some $y_n\in(x_1,x_n)$, by virtue of the Mean Value Theorem, and hence $$ \lvert\,f(x_n)\rvert\le \lvert\,f(x_1)\rvert +\lvert x_n-x_1\rvert\lvert f'(y_n)\rvert\le \lvert\,f(x_1)\rvert +2M \|f'\|_\infty. $$

  2. If If $\{x_n\}_{n\in\mathbb N}\subset\mathbb R$ is Cauchy, then $$ \lvert\,f(x_m)-f(x_n)\rvert=\lvert\,f'(y_{m,n})\rvert\lvert x_m-x_n\rvert\le\|f'\|_\infty \lvert x_m-x_n\rvert, $$ hence $\{f(x_n)\}_{n\in\mathbb N}$ Cauchy.

  3. If $x_n\to x$, then $$ \lvert\,f(x_n)-f(x)\rvert=\lvert\,f'(y_n)\rvert\lvert x_n-x\rvert\le\|f'\|_\infty \lvert x_n-x\rvert, $$ where $y\in(x,x_n)$, and hence $f(x_n)\to f(x)$.

  4. If $x,y\in\mathbb R$, then $$ \lvert\,f(x)-f(y)\rvert\le\|f'\|_\infty \lvert x-y\rvert, $$ etc...

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