Thread: Another question on sequences

I had another question:

Let x1 be greater than or equal to 2. Define inductively x(n+1) as 1+square root(xn-1) for all n. Prove that (xn) is a decreasing sequence bounded below by 2. Find the limit.

Thank you for any help. Fares.

Originally Posted by Fares23

I had another question:

Let x1 be greater than or equal to 2. Define inductively x(n+1) as 1+square root(xn-1) for all n. Prove that (xn) is a decreasing sequence bounded below by 2. Find the limit.

Thank you for any help. Fares.

The limit of I_n (nth iteration) is the same as I_{n+1} (n+1 th iteration).

Thus,
I_{n+1}=1+sqrt(I_n)
Thus,
If lim I_n= L then, lim I_{n+1}=L
Thus,
L=1+sqrt(L) (limit composition rule for sequnces)
Solve for L.

IMPORTANT. The fact that lim I_n exists was impervious to the proof. That I did not prove.

But the idea is to show that it is an increasing bounded sequence. (Use Wierestrass-Bolzano Theorom)