1. ## Sequence convergence problem

Let a1 > 1, and suppose an+1 = 2 - 1/an for all n >= 2 . Prove that (an) converges, and find its limit.

2. ## Re: Sequences convergence problem

Originally Posted by Kanwar245
Let a1 > 1, and suppose an+1 = 2 - 1/an for all n >= 2 . Prove that (an) converges, and find its limit.
It is sufficient to show that the sequence is decreasing and bounded below.
Do that with induction.

The solve the equation $\displaystyle L=2-\frac{1}{L}$.

3. ## Re: Sequences convergence problem

Don't you think there's a typo, since we are given a1 > 1, and a_n+1 is for n>= 2

4. ## Re: Sequences convergence problem

Originally Posted by Kanwar245
Don't you think there's a typo, since we are given a1 > 1, and a_n+1 is for n>= 2
Why should anyone think that?

Take a look at this

5. ## Re: Sequences convergence problem

Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that an+1 >= an right

6. ## Re: Sequences convergence problem

Originally Posted by Kanwar245
Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that an+1 >= an right
BASE CASE: Prove that $\displaystyle a_1>a_2\ge 1$.

PROVE
Inductive case: If $\displaystyle a_k>a_{k-1}\ge 1$ then $\displaystyle a_{k+1}>a_k\ge 1$.