Challenging Math.

[Jay.C]

Hey guys if you could answer the attached math question that would be amazing. I am asking on behalf of a much smarter friend. 

[keltingmeith]

The subscripts are a little confusing. Reckon you could get your friend to confirm what they are?

(Also, is this actually from a VCE class? Because that seems quite above VCE level...)

[Jay.C]

Here is the photo of the question. Also no its not from a VCE class, I just wanted to see if any of you math geniuses on ATAR notes could help! 

[wyzard]

Here is the photo of the question. Also no its not from a VCE class, I just wanted to see if any of you math geniuses on ATAR notes could help! 

This is a Real Analysis question under the topic sequences, and a really fierce one I will not bog you down with the actual proofs which require the use of really advanced maths concepts; the definition of inferior and superior limit, mathematical induction and some theorems relating to sequences. The simplest explanation is that they are the "end points" of a set of numbers.

The inferior limit is 0, which can be shown by noting that every number in the sequence is greater than or equal to zero using mathematical induction, and since is zero, the inferior limit is 0.

The exterior limit on the other hand is 1, to show this we'll need to note 2 things. Firstly, it can be shown that , so as goes to infinity, goes to 1 using the theorem . Secondly it can be shown using mathematical induction that is definitely less than 1. Hence the exterior limit is 1.