Wednesday, 4 May 2016

real analysis - Cauchy implies monotone + bounded

In my whole problem, I was stuck in one direction, which is to show (i) implies (ii).




(i) Every Cauchy sequence in R converges to a limit in R.




(ii) Every monotone and bounded sequence in R converges to a limit in R.




Explaining in english, my idea is:
Cauhcy is a bounded sequence. Given that (i) is true, i.e: this bounded cauchy sequence is convergent. Then, since not every bounded sequence is convergent, it requires some conditions so that it is convergent. One of a condition is "monotone". So, I try to use this to show (i) implies (ii). But my TA think "this is saying nothing".



If I would like to use some maths to explain, like ϵ, a sequence {xn} etc.... How could I use it??

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