How can the Monotone Convergence Theorem help? - Week 1 - Lecture 15 - Sequences and Series

[mamig]

Everyone seems to be confused about why he picked 0 and 2 in the beginning. He could have picked any 2 numbers. İf you pick -1 and 7 for example, you will see that the next number has to be between 1 and 3. Then if you do the same thing you get root3 and root5. This means that if you start with a number between -1 and 7, in your second iteration you will get a number between root3 and root5.

He just picked 0 and 2 because it shows the convergence right away.

Also he actually wrote 0<a(n+1)<2 but it's actually root2<a(n+1)<2
You can see the convergence like this easier because both upper and lower boundaries shrink.

[EduarGmez]

How do you know its less than 2, at the start?

[juniorjay]

But how do you know right at the start that it's between 0 and 2??

[zurinakasim]

use mathematical induction to prove sequence is increasing and bounded