Analysis Help

Hi, i'm quite stuck on some of these questions.
I have part solutions to some but I can't finish them.

Problem 1) I've done the first part I just can't show that lim(An) is a integer, other than noting that An is a integer sequence and is eventually constant so never exceeds the integer at which it is eventually constant at. I feel I need to write something mathematical, though.

Problem 2) I can use the Bolzano-Weierstrass Theorem to know that there is 1 convergent subsequence. I do not know how to find the other.

Problem 3) I can show a_2k, a_(2k+1) and a_3k all converge to the same limit. I do not know how to then show that a_n converges to this limit. I do not understand why the a_3k is involved, also. As surely a_3k is a subsequence where the terms are also included in a_2k and a_(2k+1).

Problem 4) I do not understand this. I just thought that if we say lim(an) = x and then by the shift rule we know lim(an+1) = x and so lim(an/(an+1)) = x/x = 1. Quite confused.

Any help would be very appreciated!

Reply 1

TSR Learn Together

Official Rep

Hi there,

While you're waiting for an answer, did you know we have 300,000 study resources that could answer your question in TSR's Learn together section?

We have everything from Teacher Marked Essays to Mindmaps and Quizzes to help you with your work. Take a look around.

If you're stuck on how to get started, try creating some resources. It's free to do and can help breakdown tough topics into manageable chunks. Get creating now.

Thanks!

Not sure what all of this is about? Head here to find out more.