1. Another question on sequences

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.

2. Originally Posted by Fares23

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)