mathematical analysis

[chriskm]

I don't understand how to prove these statements

 

[Deleted member 4993]

I don't understand how to prove these statements

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

 

[pka]

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.


View attachment 19651

The definition of cluster point is problematic for a true or false question. Although some sources will say that limit point and cluster point are the same. But not all!
The sequence \(\large a_n=\begin{cases}n &: n\text{ is odd} \\ \frac{1}{n} &: n\text{ is even}\end{cases}\)
That sequence is not bounded above. Every open interval about zero contains infinitely terms of \(a_n\) but zero is not a limit point of the sequence, but it is a cluster point of set of terms of \(a_n\).

 

[HallsofIvy]

I would say that "limit point" of a sequence and "cluster point" of a sequence are the same thing but not necessarily the same as "limit of the sequence".

 

[pka]

I would say that "limit point" of a sequence and "cluster point" of a sequence are the same thing but not necessarily the same as "limit of the sequence".

Thank you for helping to make my point. It is next to impossible to answer a true or false problem set unless exact precise set of definitions are given.