Suppose a sequence an, is defined as follows: a1 = 9/10, a2 = 10/11, an+2 = an+1an.Show that 0 < an < 1 for all (positive integers) n.

Step 1: Understanding the Problem We need to prove that for all positive integers n, the sequence an defined by a1 = 9/10, a2 = 10/11, an+2 = an+1an, satisfies 0 < an < 1.

Step 2: Base Case We can see that a1 = 9/10 and a2 = 10/11 are both between 0 and 1.

Step 3: Inductive Step Assume that for some positive integer k, ak and ak+1 are both between 0 and 1. We need to show that ak+2 = ak+1ak is also between 0 and 1.

Step 4: Proof Since 0 < ak < 1 and 0 < ak+1 < 1, their product ak+1ak is also between 0 and 1. This is because the product of two positive numbers is positive, and the product of two numbers less than 1 is less than 1.

Step 5: Conclusion Therefore, by mathematical induction, we have shown that 0 < an < 1 for all positive integers n.