Question

Nested interval property. i) Find a family {I n } of open nested intervals such that ⋂ n​ I n​ ={2 }

Answer 1

The nested interval property states that if the length of each interval approaches zero, then their intersection is a single point.

Step-by-step explanation:

The nested interval property states that for any sequence of closed and bounded intervals, if the length of each interval approaches zero, then their intersection is a single point.

In this case, we need to find a family of open nested intervals such that their intersection is the number 2.

We can define the family of open nested intervals as {In} = (2 - 1/n, 2 + 1/n), where n is a positive integer.

When n approaches infinity, both the lower and upper bounds of the intervals converge to 2, and the intersection of all the intervals is precisely the number 2.